Complement of Primitive Recursive Set

Let $S \subseteq \N$ be primitive recursive. Then its relative complement $\N \setminus S$ of $S$ in $\N$ is primitive recursive.


Intersection of Primitive Recursive Sets

Let $A, B \subseteq \N$ be subsets of the set of natural numbers $\N$. Let $A$ and $B$ both be primitive recursive. Then $A \cap B$, the intersection of $A$ and $B$, is primitive recursive.


Union of Primitive Recursive Sets

Let $A, B \subseteq \N$ be subsets of the set of natural numbers $\N$. Let $A$ and $B$ both be primitive recursive. Then $A \cup B$, the union of $A$ and $B$, is primitive recursive.


Set of Even Numbers is Primitive Recursive

Let $E \subseteq \N$ be the set of all even natural numbers. Then $E$ is primitive recursive.


Primitive Recursive Relation is URM Computable

Every primitive recursive relation is URM computable.


Equality Relation is Primitive Recursive

The relation $\operatorname{eq} \subseteq \N^2$, defined as: :$\map {\operatorname {eq} } {n, m} \iff n = m$ is primitive recursive.


Set Operations on Primitive Recursive Relations

Let $\RR_1 \subseteq N^k$ and $\RR_2 \subseteq N^k$ be $k$-ary relations on $N^k$. Let $\RR_1$ and $\RR_2$ be primitive recursive. Then the following are all primitive recursive relations: :$\TT = \neg \RR_1$ :$\UU = \RR_1 \land \RR_2$ :$\VV = \RR_1 \lor \RR_2$


Ordering Relations are Primitive Recursive

The ordering relations on $\N^2$: * $n < m$ * $n \le m$ * $n \ge m$ * $n > m$ are all primitive recursive.


Permutation of Variables of Primitive Recursive Function

Let $f: \N^k \to \N$ be a primitive recursive function. Let $\sigma$ be a permutation of $\left({1, 2, \ldots, k}\right)$. Then the function $h: \N^k \to \N$ defined as: :$h \left({n_1, n_2, \ldots, n_k}\right) = f \left({n_{\sigma \left({1}\right)}, n_{\sigma \left({2}\right)}, \ldots, n_{\sigma \left({k}\right)}}\right)$ is also primitive recursive.


Definition by Cases is Primitive Recursive

Let $\mathcal R_1, \mathcal R_2, \ldots, \mathcal R_k$ be primitive recursive relations on $\N^l$ such that: :$\forall i, j \in \set{1, 2, \ldots, k}: \mathcal R_i \implies \lnot \mathcal R_j$, that is, all relations are mutually exclusive :$\forall \tuple {n_1, n_2, \ldots, n_l} \in \N^l: \exists i \in \set {1, 2, \ldots, k}: \map {\mathcal R_i} {n_1, n_2, \ldots, n_l}$, that is, the set of relations is exhaustive. Let $\forall i \in \set {1, 2, \ldots, k}: g_i: \N^l \to \N$  be primitive recursive functions. Then the function $f: \N^l \to \N$ defined as: :$\map f {n_1, n_2, \ldots, n_l} = \begin{cases} \map {g_1} {n_1, n_2, \ldots, n_l} & : \map {\mathcal R_1} {n_1, n_2, \ldots, n_l} \\ \map {g_2} {n_1, n_2, \ldots, n_l} & : \map {\mathcal R_2} {n_1, n_2, \ldots, n_l} \\ \quad \vdots & \quad \vdots \\ \map {g_k} {n_1, n_2, \ldots, n_l} & : \map {\mathcal R_k} {n_1, n_2, \ldots, n_l} \end{cases}$ is primitive recursive.


Domain of Injection to Countable Set is Countable

Let $X$ be a set, and let $Y$ be a countable set. Let $f: X \to Y$ be an injection. Then $X$ is also countable.


Unique Code for URM Instruction

Each basic instruction $I$ in a URM Program can be identified with a unique '''code number''' $\beta \left({I}\right)$. We also define the following sets: * $\operatorname{Zinstr}$ is the set of codes of all the <tt>Zero</tt> instructions * $\operatorname{Sinstr}$ is the set of codes of all the <tt>Successor</tt> instructions * $\operatorname{Cinstr}$ is the set of codes of all the <tt>Copy</tt> instructions * $\operatorname{Jinstr}$ is the set of codes of all the <tt>Jump</tt> instructions. Then we define $\operatorname{Instr}$ to be the set of codes of all basic URM instructions. That is: :$\operatorname{Instr} = \operatorname{Zinstr} \cup \operatorname{Sinstr} \cup \operatorname{Cinstr} \cup \operatorname{Jinstr}$.


URM Instructions are Countably Infinite

The set $\Bbb I$ of all basic URM instructions is countably infinite.


URM Programs are Countably Infinite

The set $\mathbf P$ of all URM programs is countably infinite.


Unique Code for URM Program

Any URM program can be assigned a unique '''code number'''.


URM Computable Functions of One Variable is Countably Infinite

The set $\mathbf U$ of all URM computable functions of $1$ variable is countably infinite.


Natural Number Functions are Uncountable

The set of all natural number one-variable functions $\set {f: \N \to \N}$ is uncountably infinite.


Not All Natural Number Functions are Primitive Recursive

Not all functions $f: \N \to \N$ are primitive recursive.


Bounded Summation is Primitive Recursive

Let the function $f: \N^{k+1} \to \N$ be primitive recursive. Then so is the function $g: \N^{k+1} \to \N$ defined as: :$\displaystyle g \left({n_1, n_2, \ldots, n_k, z}\right) = \begin{cases} 0 & : z = 0 \\ \sum_{y \mathop = 1}^z f \left({n_1, n_2, \ldots, n_k, y}\right) & : z > 0 \end{cases}$


Bounded Product is Primitive Recursive

Let the function $f: \N^{k+1} \to \N$ be primitive recursive. Then so is the function $g: \N^{k+1} \to \N$ defined as: :$\displaystyle g \left({n_1, n_2, \ldots, n_k, z}\right) = \begin{cases} 1 & : z = 0 \\ \prod_{y=1}^z f \left({n_1, n_2, \ldots, n_k, y}\right) & : z > 0 \end{cases}$


Divisor Relation is Primitive Recursive

The divisor relation $m \divides n$ in $\N^2$ is primitive recursive.


Divisor Counting Function is Primitive Recursive

The divisor counting ($\tau$) function is primitive recursive.


Set of Prime Numbers is Primitive Recursive

The set $\Bbb P$ of prime numbers is primitive recursive.


Prime Enumeration Function is Primitive Recursive

Let the function $p: \N \to \N$ be the prime enumeration function, defined as: :$p \left({n}\right) = \begin{cases} 1 & : n = 0 \\ \mbox{the } n \mbox{th prime number} & : n > 0 \end{cases}$ Then $p$ is primitive recursive.


Length Function is Primitive Recursive

Let $n \in \N$. Let $\map \len n$ denote the length of $n$. Then the function $\len: \N \to \N$ is primitive recursive.


Prime Exponent Function is Primitive Recursive

Let $n \in \N$ be a natural number. Let $\left({n, j}\right): \N^2 \to \N$ be defined as: :$\left({n, j}\right) = \left({n}\right)_j$ where $\left({n}\right)_j$ is the prime exponent function. Then $\left({n, j}\right)$ is primitive recursive.


Set of Codes for URM Instructions is Primitive Recursive

The set $\operatorname{Instr}$ of codes of all basic URM instructions is primitive recursive.


Set of Sequence Codes is Primitive Recursive

Let $\operatorname{Seq}$ be the set of all code numbers of finite sequences in $\N$. Then $\operatorname{Seq}$ is primitive recursive.


Set of Codes for URM Programs is Primitive Recursive

Let $\operatorname{Prog}$ be the set of all code numbers of URM programs. Then $\operatorname{Prog}$ is a primitive recursive set.


Upper Bounds for Prime Numbers

Let $p: \N \to \N$ be the prime enumeration function. Then $\forall n \in \N$, the value of $\map p n$ is bounded above. In particular:


Minimization on Relation Equivalent to Minimization on Function

Let $\mathcal R$ be a $k+1$-ary relation on $\N^{k+1}$. Then the function $g: \N^{k+1} \to \N$ defined as: :$g \left({n_1, n_2, \ldots, n_k, z}\right) = \mu y \ \mathcal R \left({n_1, n_2, \ldots, n_k, y}\right)$ where $\mu y \ \mathcal R \left({n_1, n_2, \ldots, n_k, y}\right)$ is the minimization operation on $\mathcal R$ is equivalent to minimization on a total function.


Function Obtained by Minimization from URM Computable Functions

Let the function $f: \N^{k+1} \to \N$ be a URM computable function. Let $g: \N^k \to \N$ be the function obtained by minimization from $f$ thus: :$g \left({n_1, n_2, \ldots, n_k}\right) \approx \mu y \left({f \left({n_1, n_2, \ldots, n_k}\right) = 0}\right)$. Then $g$ is also URM computable.


Function Obtained by Minimization from URM Computable Relations

Let $\mathcal R$ be a URM computable $k+1$-ary relation on $\N^{k+1}$. Let the function $f: \N^{k+1} \to \N$ be a URM computable function. Let $g: \N^k \to \N$ be the function obtained by minimization from $f$ thus: :$g \left({n_1, n_2, \ldots, n_k}\right) \approx \mu y \mathcal R \left({n_1, n_2, \ldots, n_k, y}\right)$ Then $g$ is also URM computable.


Recursive Function is URM Computable

Every recursive function is URM computable.


Unique Code for State of URM Program

Every state of a URM program can be assigned a unique '''code number'''. This code number is called the '''state code''' (or '''situation code''').


State Code Function is Primitive Recursive

Let $k \in \N^*$. Let $e = \gamma \left({P}\right)$ be the code number of a URM program $P$. Let $\left({n_1, n_2, \ldots, n_k}\right)$ be the input of $P$. Let $S_k: \N^{k+2} \to \N$ be the function defined as: :$S_k \left({e, n_1, n_2, \ldots, n_k, t}\right)$ is the state code for $P$ at stage $t$ of computation of $P$. If $e$ does not code a URM Program then $S_k = 0$. Also, if $P$ terminates at stage $t_0$, then we put: :$\forall t > t_0: S_k \left({e, n_1, n_2, \ldots, n_k, t}\right) = S_k \left({e, n_1, n_2, \ldots, n_k, t_0}\right)$. Then for all $k \ge 1$, the function $S_k$ is primitive recursive.


URM Computable Function is Recursive

Every URM computable function is recursive.


Kleene's Normal Form Theorem

For each integer $k \ge 1$, there exists: * a primitive recursive $k+1$-ary relation $T_k$; * a primitive recursive function $U: \N \to \N$ such that a partial function $f: \N^k \to \N$ is recursive iff, for some $e \in \N$ and all $\left({n_1, n_2, \ldots, n_k}\right) \in \N^k$: :$f \left({n_1, n_2, \ldots, n_k}\right) \approx U \left({\mu z \ T_k \left({e, n_1, n_2, \ldots, n_k, z}\right)}\right)$


Recursive Function uses One Minimization

Every recursive function can be obtained from the basic primitive recursive functions using: * substitution * primitive recursion * at most one minimization on a function.


Universal URM Computable Functions

For each integer $k \ge 1$, there exists a URM computable function: :$\Phi_k: \N^{k+1} \to \N$ such that for each URM computable function $f: \N^k \to \N$ there exists a natural number $e$ such that: :$\forall \left({n_1, n_2, \ldots, n_k}\right) \in \N^k: f \left({n_1, n_2, \ldots, n_k}\right) \approx \Phi_k \left({e, n_1, n_2, \ldots, n_k}\right)$. This function $\Phi_k$ is '''universal for URM computable functions of $k$ variables'''.


Universal URM Programs

For each integer $k \ge 1$, there exists a URM program $P_k$ such that: For each URM program $P$ there exists a natural number $e$ such that: For all $\left({n_1, n_2, \ldots, n_k}\right) \in \N^k$, the computation using the program $P_k$ with input $\left({e, n_1, n_2, \ldots, n_k}\right)$ has the same output as the computation using the program $P$ with input $\left({n_1, n_2, \ldots, n_k}\right)$. This function $P_k$ is a '''universal program for URM computations with $k$ inputs'''.


Combination of Recursive Functions

Let $f: \N^k \to \N$ and $g: \N^k \to \N$ be recursive functions (not necessarily total), where $k \ge 1$. Let $\mathcal R$ be a $k$-ary relation such that: : if $\mathcal R \left({n_1, n_2, \ldots, n_k}\right)$ holds, then $f \left({n_1, n_2, \ldots, n_k}\right)$ is defined : if $\mathcal R \left({n_1, n_2, \ldots, n_k}\right)$ does not hold, then $g \left({n_1, n_2, \ldots, n_k}\right)$ is defined. Let $h: \N^k \to \N$ be the function defined as: :$h \left({n_1, n_2, \ldots, n_k}\right) = \begin{cases} f \left({n_1, n_2, \ldots, n_k}\right) & : \text{if } \mathcal R \left({n_1, n_2, \ldots, n_k}\right) \text { holds} \\ g \left({n_1, n_2, \ldots, n_k}\right) & : \text{otherwise} \end{cases}$ so that $h$ is total. Then $h$ is recursive.


Not All URM Computable Functions are Primitive Recursive

There exist URM computable functions which are not primitive recursive.


Cantor's Diagonal Argument

Let $S$ be a set such that $\card S > 1$, that is, such that $S$ is not a singleton. Let $\mathbb F$ be the set of all mappings from the natural numbers $\N$ to $S$: :$\mathbb F = \set {f: \N \to S}$ Then $\mathbb F$ is uncountably infinite.


Infinite if Injection from Natural Numbers

Let $S$ be a set. Let there exist an injection $\phi: \N \to S$ from the natural numbers to $S$. Then $S$ is infinite.


Halting Problem is Not Algorithmically Decidable

Let $H: \N^2 \to \N$ be the function given by: :$\map H {m, n} = 1$ if $m$ codes a URM program which halts with input $n$ :$\map H {m, n} = 0$ otherwise. Then $H$ is not recursive.


Set of Total Functions is Not Recursive

The set $\operatorname{Tot}$ of natural numbers which code URM programs which compute total functions of one variable is not recursive.


Infinitely Many Programs for URM Computable Function

Let $g: \N^k \to \N$ be a URM computable function. Then there is an infinite number of URM programs which compute $g$.


Bounds of GCD for Sum and Difference Congruent Squares

Let $x, y, n$ be integers. Let: :$x \not \equiv \pm y \pmod n$ and: :$x^2 \equiv y^2 \pmod n$ where $a \equiv b \pmod n$ denotes that $a$ is congruent to $b$ modulo $n$. Then: :$1 < \gcd \set {x - y, n} < n$ and: :$1 < \gcd \set {x + y, n} < n$ where $\gcd \set {a, b}$ is the GCD of $a$ and $b$.


Triangle Angle-Side-Angle Equality

If two triangles have: :two angles equal to two angles, respectively :the sides between the two angles equal then the remaining angles are equal, and the remaining sides equal the respective sides. That is to say, if two pairs of angles and the included sides are equal, then the triangles are congruent.


Triangle Side-Angle-Angle Equality

If two triangles have: :two angles equal to two angles, respectively :the sides opposite one pair of equal angles equal then the remaining angles are equal, and the remaining sides equal the respective sides. That is to say, if two pairs of angles and a pair of opposite sides are equal, then the triangles are congruent.


Equal Alternate Angles implies Parallel Lines

Given two infinite straight lines which are cut by a transversal, if the alternate angles are equal, then the lines are parallel. {{:Euclid:Proposition/I/27}}


Parallelism is Transitive Relation

Parallelism between straight lines is a transitive relation. {{:Euclid:Proposition/I/30}}


Sum of Sequence of Products of Consecutive Integers

{{begin-eqn}} {{eqn | l = \sum_{j \mathop = 1}^n j \paren {j + 1}       | r = 1 \times 2 + 2 \times 3 + \dotsb + n \paren {n + 1}       | c =  }} {{eqn | r = \frac {n \paren {n + 1} \paren {n + 2} } 3       | c =  }} {{end-eqn}}


Sum of Sequence of Products of Consecutive Reciprocals

:$\displaystyle \sum_{j \mathop = 1}^n \frac 1 {j \paren {j +1} } = \frac n {n + 1}$


Sum of Sequence of Fibonacci Numbers

:$\displaystyle \forall n \in \Z_{\ge 0}: \sum_{j \mathop = 0}^n F_j = F_{n + 2} - 1$


Sum of Sequence of Even Index Fibonacci Numbers

{{begin-eqn}} {{eqn | lo= \forall n \ge 1:       | l = \sum_{j \mathop = 1}^n F_{2 j}       | r = F_2 + F_4 + F_6 + \cdots + F_{2 n}       | c =  }} {{eqn | r = F_{2 n + 1} - 1       | c =  }} {{end-eqn}}


Sum of Odd Sequence of Products of Consecutive Fibonacci Numbers

:$\ds \sum_{j \mathop = 1}^{2 n - 1} F_j F_{j + 1} = {F_{2 n} }^2$


Lucas Number as Sum of Fibonacci Numbers

Let $L_k$ be the $k$th Lucas number, defined as: :$L_n = \begin{cases} 2 & : n = 0 \\ 1 & : n = 1 \\ L_{n - 1} + L_{n - 2} & : \text{otherwise} \end{cases}$ Then: :$L_n = F_{n - 1} + F_{n + 1}$


Sum of Sequence of Product of Lucas Numbers with Powers of 2

Let $L_k$ be the $k$th Lucas number. Let $F_k$ be the $k$th Fibonacci number. Then: :$\displaystyle \forall n \in \N_{>0}: \sum_{j \mathop = 1}^n 2^{j - 1} L_j = 2^n F_{n + 1} - 1$ That is: :$2^0 L_1 + 2^1 L_2 + 2^2 L_3 + \cdots + 2^{n - 1} L^n = 2^n F_{n + 1} - 1$


Sum of Odd Positive Powers

Let $n \in \N$ be an odd positive integer. Let $x, y \in \Z_{>0}$ be (strictly) positive integers. Then $x + y$ is a divisor of $x^n + y^n$.


Condition for Well-Foundedness

Let $\struct {S, \preceq}$ be an ordered set. Then $\struct {S, \preceq}$ is well-founded {{iff}} there is no infinite sequence $\sequence {a_n}$ of elements of $S$ such that: :$\forall n \in \N: a_{n + 1} \prec a_n$ That is, {{iff}} there is no infinite sequence $\sequence {a_n}$ such that $a_0 \succ a_1 \succ a_2 \succ \cdots$.


Subset Relation on Power Set is Partial Ordering

Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $\struct {\powerset S, \subseteq}$ be the relational structure defined on $\powerset S$ by the relation $\subseteq$. Then $\struct {\powerset S, \subseteq}$ is an ordered set. The ordering $\subseteq$ is partial {{iff}} $S$ is neither empty nor a singleton; otherwise it is total.


Ordering is Equivalent to Subset Relation

Let $\struct {S, \preceq}$ be an ordered set. Then there exists a set $\mathbb S$ of subsets of $S$ such that: :$\struct {S, \preceq} \cong \struct {\mathbb S, \subseteq}$ where: :$\struct {\mathbb S, \subseteq}$ is the relational structure consisting of $\mathbb S$ and the subset relation :$\cong$ denotes order isomorphism. Hence any ordering on a set can be modelled uniquely by a set of subsets of that set under the subset relation.


Subset Relation is Ordering

Let $\mathbb S$ be a set of sets. Then $\subseteq$ is an ordering on $\mathbb S$. In other words, let $\struct {\mathbb S, \subseteq}$ be the relational structure defined on $\mathbb S$ by the relation $\subseteq$. Then $\struct {\mathbb S, \subseteq}$ is an ordered set.


Principle of Mathematical Induction/Well-Ordered Set

Let $\struct {S, \preceq}$ be a well-ordered set. Let $T \subseteq S$ be a subset of $S$ such that: :$\forall s \in S: \paren {\forall t \in S: t \prec s \implies t \in T} \implies s \in T$ Then $T = S$.


Order Isomorphism from Woset onto Subset

Let $\struct {S, \preceq}$ be a woset. Let $T \subseteq S$. Let $f: S \to T$ be an order isomorphism. Then $\forall x \in S: x \preceq \map f x$.


Order Isomorphism between Wosets is Unique

Let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be wosets. Let $\struct {S_1, \preceq_1} \cong \struct {S_2, \preceq_2}$, that is, let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be order isomorphic. Then there is exactly one mapping $f: S_1 \to S_2$ such that $f$ is an order isomorphism.


No Isomorphism from Woset to Initial Segment

Let $\struct {S, \preceq}$ be a woset. Let $a \in S$, and let $S_a$ be the initial segment of $S$ determined by $a$. Then there is no order isomorphism between $S$ and $S_a$.


Woset is Isomorphic to Set of its Initial Segments

Let $\struct {S, \preceq}$ be a well-ordered set. Let: :$A = \set {a^\prec: a \in S}$ where $a^\prec$ is the strict lower closure of $S$ determined by $a$. Then: :$\struct {S, \preceq} \cong \struct {A, \subseteq}$ where $\cong$ denotes order isomorphism.


Ordering on Ordinal is Subset Relation

Let $\struct {S, \prec}$ be an ordinal. Then $\forall x, y \in S:$ :$x \in y \iff x \prec y \iff S_x \subsetneqq S_y \iff x \subsetneqq y$ where $S_x$ and $S_y$ are the initial segments of $S$ determined by $x$ and $y$ respectively. Thus there is no need to specify what the ordering on an ordinal is -- it is always the subset relation.


Initial Segment of Ordinal is Ordinal

Let $S$ be an ordinal, and suppose that $a \in S$. Then the initial segment $S_a = a$ of $S$ determined by $a$ is also an ordinal. In other words, every element of an ordinal is also an ordinal.


Ordinal Subset of Ordinal is Initial Segment

Let $S$ be an ordinal. Let $T \subset S$ also be an ordinal. Then $\exists a \in S: T = S_a$, where $S_a$ is the initial segment of $S$ determined by $a$. That is, $T = S_a = a \in S$.


Intersection of Two Ordinals is Ordinal

Let $S, T$ be ordinals. Then $S \cap T$ is an ordinal.


Relation between Two Ordinals

Let $S$ and $T$ be ordinals. Then either $S \subseteq T$ or $T \subseteq S$.


Isomorphic Ordinals are Equal

Let $A$ and $B$ be ordinals that are order isomorphic. Then $A = B$.


Condition for Woset to be Isomorphic to Ordinal

Let $\struct {S, \preceq}$ be a woset. Let $\struct {S, \preceq}$ be such that $\forall a \in S$, the initial segment $S_a$ of $S$ determined by $a$ is order isomorphic to some ordinal. Then $\struct {S, \preceq}$ itself is order isomorphic to an ordinal.


Woset is Isomorphic to Unique Ordinal

Every woset is order isomorphic to a unique ordinal.


Product of the Incidence Matrix of a BIBD with its Transpose

For any BIBD with parameters $v,k,\lambda$, let $A$ be its block incidence matrix. Then: : $A^\intercal \cdot A = \left({a_{ij}}\right) = \left({r - \lambda}\right) I_v + \lambda J_v$ where: * $A$ is $v \times b$ * $A^\intercal$ is the transpose of $A$ * $J_v$ is the all $v\times{v}$ $ 1$'s matrix * $I_v$ is the $v\times{v}$ identity matrix. That is: : $A^\intercal \cdot A = \begin{bmatrix} r & \lambda & \cdots & \lambda \\ \lambda & r & \cdots & \lambda \\ \vdots & \vdots & \ddots & \vdots \\ \lambda & \lambda & \cdots & r \\ \end{bmatrix}$


Fisher's Inequality

For any BIBD $\struct {v, k, \lambda}$, the number of blocks $b$ must be greater then or equal to the number of points $v$: :$ b \ge v$


Self-Distributive Law for Conditional

The following is known as the Self-Distributive Law:


Disjunction of Conditional and Converse

: $\vdash \left({p \implies q}\right) \lor \left({q \implies p}\right)$


Characteristics of Eulerian Graph

A finite (undirected) graph is Eulerian {{iff}} it is connected and each vertex is even. Note that the definition of graph here includes: * Simple graph * Loop-graph * Multigraph * Loop-multigraph but does not include directed graph.


De Morgan's Laws

'''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:


Four Color Theorem

Any planar graph $G$ can be assigned a proper vertex $k$-coloring such that $k \le 4$.


Dirac's Theorem

If a connected graph $G$ has $n \ge 3$ vertices and the degree of each vertex is at least $\dfrac n 2$, then $G$ is Hamiltonian.


Functionally Complete Logical Connectives

These sets of logical connectives are functionally complete:


Properties of NAND

Let $\uparrow$ signify the NAND operation. The following results hold:


Properties of NOR

Let $\downarrow$ signify the NOR operation. The following results hold:


NAND and NOR are Functionally Complete

The NAND and NOR operators are each functionally complete. That is, NAND and NOR are Sheffer operators.


Equivalence of Semantic Consequence and Logical Implication

Let $U = \set {\phi_1, \phi_2, \ldots, \phi_m, \ldots}$ be a countable set of propositional formulas. Let $\psi$ be a propositional formula. Then $U \models \psi$ {{iff}} $U \vdash \psi$. That is, semantic consequence is equivalent to provable consequence.


WFFs of PropLog of Length 1

The only WFFs of propositional logic of length $1$ are: * The letters of the formal grammar of propositional logic $\mathcal L_0$ * The tautology symbol $\top$ * The contradiction symbol $\bot$.


WFF of PropLog is Balanced

Let $\mathbf A$ be a WFF of propositional logic. Then $\mathbf A$ is a balanced string.


Initial Part of WFF of PropLog is not WFF

Let $\mathbf A$ be a WFF of propositional logic. Let $\mathbf S$ be an initial part of $\mathbf A$. Then $\mathbf S$ is not a WFF of propositional logic.


Construction of Parallel Line

Given a straight line, and a given point not on that straight line, it is possible to draw a parallel to the given straight line. {{:Euclid:Proposition/I/31}}


Sum of Angles of Triangle equals Two Right Angles

In a triangle, the sum of the three interior angles equals two right angles. {{:Euclid:Proposition/I/32}}


Equivalence of Logical Implication and Conditional

: $\left({p \implies q}\right) \dashv \vdash \left({p \vdash q}\right)$ That is, the conditional is logically equivalent to logical implication.


Models for Propositional Logic

This page gathers together some useful results that can be used in the derivation of proofs by propositional tableau. Let $\mathcal M$ be a model for propositional logic, and let $\mathbf A$ and $\mathbf B$ be WFFs of propositional logic. Then the following results hold. The symbol $\models$ is used throughout to denote semantic consequence.


No Boolean Interpretation Models a WFF and its Negation

Let $v$ be a boolean interpretation. Let $\mathbf A$ be a WFF of propositional logic. Then $v$ can not model both $\mathbf A$ and $\neg \mathbf A$.


Extended Soundness Theorem for Propositional Tableaus and Boolean Interpretations

Tableau proofs (in terms of propositional tableaus) are a strongly sound proof system for boolean interpretations. That is, for every collection $\mathbf H$ of WFFs of propositional logic and every WFF $\mathbf A$: :$\mathbf H \vdash_{\mathrm{PT}} \mathbf A$ implies $\mathbf H \models_{\mathrm{BI}} \mathbf A$


Extended Completeness Theorem for Propositional Tableaus and Boolean Interpretations

Tableau proofs (in terms of propositional tableaus) are a strongly complete proof system for boolean interpretations. More precisely, for every countable collection $\mathbf H$ of WFFs of propositional logic and every WFF $\mathbf A$: :$\mathbf H \models_{\mathrm{BI}} \mathbf A$ implies $\mathbf H \vdash_{\mathrm{PT}} \mathbf A$


Completeness Theorem for Propositional Tableaus and Boolean Interpretations

Tableau proofs (in terms of propositional tableaus) are a complete proof system for boolean interpretations. That is, for every WFF $\mathbf A$: :$\models_{\mathrm{BI}} \mathbf A$ implies $\vdash_{\mathrm{PT}} \mathbf A$


König's Tree Lemma

Let $T$ be a rooted tree with an infinite number of nodes, each with a finite number of children. Then $T$ has a branch of infinite length.


Compactness Theorem for Boolean Interpretations

Let $\mathbf H$ be a countable set of WFFs of propositional logic. Suppose $\mathbf H$ is finitely satisfiable for boolean interpretations. That is, suppose that every finite subset $\mathbf H' \subseteq \mathbf H$ is satisfiable for boolean interpretations. Then $\mathbf H$ has a model.


De Morgan's Laws (Predicate Logic)

{{:De Morgan's Laws (Predicate Logic)/Assertion of Universality}}


Initial Part of WFF of Predicate Logic is not WFF

Let $\mathbf A$ be a WFF of predicate logic. Let $\mathbf S$ be an initial part of $\mathbf A$. Then $\mathbf S$ is not a WFF of predicate logic.


Unique Readability Theorem of Predicate Calculus

Each WFF of predicate logic which starts with a left bracket or a negation sign has exactly one main connective. {{Explain|probably, in the correct formulation of 'main connective', the quantifiers ought to be connectives as well}}


Quantifier has Unique Scope

Let $\mathbf A$ be a WFF of predicate logic. Let $Q$ be an occurrence of a quantifier in $\mathbf A$. Then there exists a unique well-formed part of $\mathbf A$ which (omitting outermost parentheses) begins with that occurrence $Q$. This unique well-formed part of $\mathbf A$ is called the scope of the occurrence of $Q$.


Restriction of Operation Distributivity

Let $\left({S, *, \circ}\right)$ be an algebraic structure. Let $T \subseteq S$. If the operation $\circ$ is distributive over $*$ in $\left({S, *, \circ}\right)$, then it is also distributive over $*$ on a restriction $\left({T, * \restriction_T, \circ \restriction_T}\right)$.


Union of Intersections

:$\paren {S_1 \cap S_2} \cup \paren {T_1 \cap T_2} \subseteq S_1 \cup T_1$


Union of Symmetric Differences

Let $R, S, T$ be sets. Then: :$\left({R * S}\right) \cup \left({S * T}\right) = \left({R \cup S \cup T}\right) \setminus \left({R \cap S \cap T}\right)$ where $R * S$ denotes the symmetric difference between $R$ and $S$.


Difference of Unions is Subset of Union of Differences

Let $I$ be an indexing set. Let $S_\alpha, T_\alpha$ be sets, for all $\alpha \in I$. Then: :$\displaystyle \paren {\bigcup_{\alpha \mathop \in I} S_\alpha} \setminus \paren {\bigcup_{\alpha \mathop \in I} T_\alpha} \subseteq \bigcup_{\alpha \mathop \in I} \paren {S_\alpha \setminus T_\alpha}$ where $S_\alpha \setminus T_\alpha$ denotes set difference.


Symmetric Difference of Unions is Subset of Union of Symmetric Differences

Let $I$ be an indexing set. Let $S_\alpha, T_\alpha$ be sets, for all $\alpha \in I$. Then: :$\displaystyle \bigcup_{\alpha \mathop \in I} S_\alpha * \bigcup_{\alpha \mathop \in I} T_\alpha \subseteq \bigcup_{\alpha \mathop \in I} \paren {S_\alpha * T_\alpha}$ where $S * T$ is the symmetric difference between $S$ and $T$.


Pseudometric Defines an Equivalence Relation

Let $X$ be a set on which there is a pseudometric $d: X \times X \to \R$. For any $x, y \in X$, let $x \sim y$ denote that $d \left({x, y}\right) = 0$. Then $\sim$ is an equivalence relation, and the equivalence classes consist of sets of elements of $X$ at zero distance from each other.


First-Order Reaction

Let a substance decompose spontaneously in a '''first-order reaction'''. Let $x_0$ be a measure of the quantity of that substance at time $t = 0$. Let the quantity of the substance that remains after time $t$ be $x$. Then: :$x = x_0 e^{-k t}$ where $k$ is the rate constant.


Solution to First Order ODE

Let: :$\displaystyle \Phi = \frac {\d y} {\d x} = \map f {x, y}$ be a first order ordinary differential eqn. Then $\Phi$ has a general solution which can be expressed in terms of an indefinite integral of $\map f x$: :$\displaystyle y = \int \map f {x, y} \rd x + C$ where $C$ is an arbitrary constant.


Solution to First Order Initial Value Problem

Let $\map y x$ be a solution to the first order ordinary differential equation: :$\dfrac {\d y} {\d x} = \map f {x, y}$ which is subject to an initial condition: $\tuple {a, b}$. Then this problem is equivalent to the integral equation: :$\displaystyle y = b + \int_a^x \map f {t, \map y t} \rd t$


Condition for Lipschitz Condition to be Satisfied

Let $f$ be a real function. Then $f$ satisfies the Lipschitz condition on a closed real interval $\closedint a b$ if: :$\forall y \in \closedint a b: \exists A \in \R: \size {\map {\phi'} y} \le A$


Separation of Variables

Suppose a first order ordinary differential equation can be expressible in this form: :$\dfrac {\d y} {\d x} = \map g x \, \map h y$ Then the equation is said to '''have separable variables''', or '''be separable'''. Its general solution is found by solving the integration: :$\displaystyle \int \frac {\d y} {\map h y} = \int \map g x \rd x + C$


Quotient of Homogeneous Functions

Let $\map M {x, y}$ and $\map N {x, y}$ be homogeneous functions of the same degree. Then: :$\dfrac {\map M {x, y} } {\map N {x, y} }$ is homogeneous of zero degree.


Solution to Homogeneous Differential Equation

Let: :$\map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$ be a homogeneous differential equation. It can be solved by making the substitution $z = \dfrac y x$. Its solution is: :$\displaystyle \ln x = \int \frac {\d z} {\map f {1, z} - z} + C$ where: :$\map f {x, y} = -\dfrac {\map M {x, y} } {\map N {x, y} }$


Solution to Exact Differential Equation

The first order ordinary differential equation: :$F = \map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$ is an exact differential equation {{iff}}: :$\dfrac {\partial M} {\partial y} = \dfrac {\partial N} {\partial x}$ The general solution of such an equation is: :$\map f {x, y} = C$ where: :$\dfrac {\partial f} {\partial x} = M$ :$\dfrac {\partial f} {\partial y} = N$


Existence of Integrating Factor

Let the first order ordinary differential equation: :$(1): \quad M \left({x, y}\right) + N \left({x, y}\right) \dfrac {\mathrm d y} {\mathrm d x} = 0$ be such that $M$ and $N$ are real functions of two variables which are ''not'' homogeneous functions of the same degree. Suppose also that: :$\dfrac {\partial M} {\partial y} \ne \dfrac {\partial N} {\partial x}$ that is, $(1)$ is not exact. Finally, suppose that $(1)$ has a general solution. Then it is always possible to find an integrating factor $\mu \left({x, y}\right)$ such that: :$\mu \left({x, y}\right) \left({M \left({x, y}\right) + N \left({x, y}\right) \dfrac {\mathrm d y} {\mathrm d x}}\right) = 0$ is an exact differential equation. Hence it is possible to find that solution by Solution to Exact Differential Equation.


Integrating Factor for First Order ODE

Let the first order ordinary differential equation: :$(1): \quad \map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$ be non-homogeneous and not exact. By Existence of Integrating Factor, if $(1)$ has a general solution, there exists an integrating factor $\map \mu {x, y}$ such that: :$\displaystyle \map \mu {x, y} \paren {\map M {x, y} + \map N {x, y} \frac {\d y} {\d x} } = 0$ is an exact differential equation. Unfortunately, there is no systematic method of finding such a $\map \mu {x, y}$ for all such equations $(1)$.<ref>"In general this is quite difficult." ::: -- {{BookReference|Differential Equations|1972|George F. Simmons}}: $\S 2.9$: Integrating Factors</ref> However, there are certain types of first order ODE for which an integrating factor ''can'' be found procedurally.


Solution to Linear First Order Ordinary Differential Equation

A linear first order ordinary differential equation in the form: :$\dfrac {\d y} {\d x} + \map P x y = \map Q x$ has the general solution: :$\ds y = e^{-\int P \rd x} \paren {\int Q e^{\int P \rd x} \rd x + C}$


Extremal Length of Composition

Let $\Gamma_1$ and $\Gamma_2$ be families of (unions of) rectifiable curves on a Riemann surface $X$. Let $\Gamma_1$ and $\Gamma_2$ be disjoint in the sense that there exist disjoint Borel sets $E_1 \subseteq X$ and $E_2 \subseteq X$ with $\bigcup \Gamma_1 \subset E_1$ and $\bigcup \Gamma_2 \subset E_2$. The extremal length of the family: :$\Gamma := \left\{{\gamma_1 \cup \gamma_2:\ \gamma_1 \in \Gamma_1 \text{ and }\gamma_2 \in \Gamma_2}\right\}$ satisfies: :$\lambda \left({\Gamma}\right) = \lambda \left({\Gamma_1}\right) + \lambda \left({\Gamma_2}\right)$


Squares with All Odd Digits

The only squares whose digits<ref>That is, when written in base 10 notation.</ref> are all odd are $1$ and $9$.


Parity of Integer equals Parity of its Square

Let $p \in \Z$ be an integer. Then $p$ is even {{iff}} $p^2$ is even.


Weierstrass M-Test

Let $f_n$ be a sequence of real functions defined on a domain $D \subseteq \R$. Let $\displaystyle \sup_{x \mathop \in D} \size {\map {f_n} x} \le M_n$ for each integer $n$ and some constants $M_n$ Let $\displaystyle \sum_{i \mathop = 1}^\infty M_i < \infty$. Then $\displaystyle \sum_{i \mathop = 1}^\infty f_i$ converges uniformly on $D$.


Measurable Sets form Sigma-Algebra

Let $\mu^*$ be an outer measure on a set $X$. Then the set $\map {\mathfrak M} {\mu^*}$ of all $\mu^*$-measurable subsets of $X$ is a $\sigma$-algebra.


Markov's Inequality

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $f$ be an $A$-measurable function where $A \in \Sigma$. Then: :$\displaystyle \mu \left({\left\{ {x \in A: \left\vert{f \left({x}\right)}\right\vert \ge t}\right\} }\right) \le \frac 1 t \int_A \left\vert{f}\right\vert \mathrm d \mu$ for any positive $t \in \R$.


Limit Superior includes Limit Inferior

Let $\set {E_n : n \in \N}$ be a sequence of sets. Then: :$\ds \liminf_{n \mathop \to \infty} E_n \subseteq \limsup_{n \mathop \to \infty} E_n$


Construction of Outer Measure

{{questionable|The sense of this seems to have got lost at around the time of the edit of 25th March 2012. Presumably the intention was to extend this definition from the set of mappings on the powerset of $X$ to an arbitrary subset of them -- but the definition of $\mu^*$ seems to be inconsistent with this. Either a more rigorous explanation of exactly what all the concepts are here, or the theorem has been mis-stated.}} Let $X$ be a set. Let $\powerset X$ be the power set of $X$. Let $\AA$ be a subset of $\powerset X$ which contains the empty set. Let $\overline \R_{\ge 0}$ denote the set of positive extended real numbers. Let $\gamma: \AA \to \overline \R_{\ge 0}$ be a mapping such that $\map \gamma \O = 0$. Let $\mu^*: \AA \to \overline \R_{\ge 0}$ be the mapping defined as: :$\displaystyle \forall S \in \powerset X: \map {\mu^*} S = \inf \set {\sum_{n \mathop = 0}^\infty \map \gamma {A_n}: \forall n \in \N : A_n \in \AA, \ S \subseteq \bigcup_{n \mathop = 0}^\infty A_n}$ Then $\mu^*$ is an outer measure on $X$. The infimum of the empty set is the greatest element, $+\infty$.


Additive and Countably Subadditive Function is Countably Additive

Let $\Sigma$ be a $\sigma$-algebra over a set $X$. Let $f: \Sigma \to \overline \R_{\ge 0}$ be an additive and countably subadditive function, where $\overline \R_{\ge 0}$ denotes the set of positive extended real numbers. Then $f$ is countably additive.


Banach Fixed-Point Theorem

Let $\struct {M, d}$ be a complete metric space. Let $f: M \to M$ be a contraction. That is, there exists $q \in \hointr 0 1$ such that for all $x, y \in M$: :$\map d {\map f x, \map f y} \le q \, \map d {x, y}$ Then there exists a unique fixed point of $f$.


Measure of Empty Set is Zero

Let $\struct {X, \Sigma, \mu}$ be a measure space. Then $\map \mu \O = 0$. That is, $\O$ is a $\mu$-null set.


Convergence a.u. Implies Convergence a.e.

Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $f_n: D \to \R$ be a sequence of $\Sigma$-measurable functions for $D \in \Sigma$. Let $f_n$ converge a.u. to a function $f$ on $D$. Then $f_n$ converges a.e. to $f$.


Existence of Solution to System of First Order ODEs

Consider the system of initial value problems: :$\begin{cases} \dfrac {\mathrm d y} {\mathrm d x} = f \left({x, y, z}\right) & : y \left({x_0}\right) = y_0 \\ & \\ \dfrac {\mathrm d z} {\mathrm d x} = g \left({x, y, z}\right) & : z \left({x_0}\right) = z_0 \\ \end{cases}$ where $f \left({x, y, z}\right)$ and $g \left({x, y, z}\right)$ are continuous real functions in some region of space $x y z$ that contains the point $\left({x_0, y_0, z_0}\right)$. Then this system of equations has a unique solution which exists on some interval $\left|{x - x_0}\right| \le h$.


Little Bézout Theorem

Let $\map {P_n} x$ be a polynomial of degree $n$ in $x$. Let $a$ be a constant. Then the remainder of $\map {P_n} x$ when divided by $x - a$ is equal to $\map {P_n} a$.


Thales' Theorem

Let $A$ and $B$ be two points on opposite ends of the diameter of a circle. Let $C$ be another point on the circle such that $C \ne A, B$. Then the lines $AC$ and $BC$ are perpendicular to each other. :400px


Edge of Polyhedron has no Curvature

The edge of a polyhedron has zero curvature.


Properties of Dot Product

Let $\mathbf u, \mathbf v, \mathbf w$ be vectors in the vector space $\R^n$. Let $c$ be a real scalar. The dot product has the following properties:


Dot Product is Inner Product

The dot product is an inner product.


Factor Matrix in the Inner Product

:$\left \langle {A \mathbf u,\mathbf v} \right \rangle = \left \langle {\mathbf u, A^T\mathbf v} \right \rangle$ where $\mathbf u$ and $\mathbf v$ are both $1 \times n$ column vectors.


Empty Set Disjoint with Itself

The empty set is disjoint with itself: :$\varnothing \cap \varnothing = \varnothing$


Pi is Irrational

Pi ($\pi$) is irrational.


Properties of Hadamard Product

Let $\map {\MM_S} {m, n}$ be a $m \times n$ matrix space over $S$ over an algebraic structure $\struct {S, \cdot}$. Let $\mathbf A, \mathbf B \in \map {\MM_S} {m, n}$. Let $\mathbf A \circ \mathbf B$ be defined as the Hadamard product of $\mathbf A$ and $\mathbf B$. The operation $\circ$ of Hadamard product satisfies the following properties: :$\circ$ is closed on $\map {\MM_S} {m, n}$ {{iff}} $\cdot$ is closed on $\struct {S, \cdot}$ :$\circ$ is associative on $\map {\MM_S} {m, n}$ {{iff}} $\cdot$ is associative on $\struct {S, \cdot}$ :$\circ$ is commutative on $\map {\MM_S} {m, n}$ {{iff}} $\cdot$ is commutative on $\struct {S, \cdot}$.


Graph Isomorphism is Equivalence Relation

Graph isomorphism is an equivalence relation.


Relation Isomorphism is Equivalence Relation

Relation isomorphism is an equivalence relation.


Vertex Condition for Isomorphic Graphs

Let $G_1$ and $G_2$ be isomorphic graphs. Then the degrees of the vertices of $G_1$ are exactly the same as the degrees of the vertices of $G_2$.


Pigeonhole Principle

Let $S$ be a finite set whose cardinality is $n$. Let $S_1, S_2, \ldots, S_k$ be a partition of $S$ into $k$ subsets. Then: :at least one subset $S_i$ of $S$ contains at least $\ceiling {\dfrac n k}$ elements where $\ceiling {\, \cdot \,}$ denotes the ceiling function.


Graph Connectedness is Equivalence Relation

Let $G = \struct {V, E}$ be a graph. Let $\to$ denote the relation '''is connected to''' on the set $V$. Then $\to$ is an equivalence relation.


Condition for Edge to be Bridge

Let $G = \struct {V, E}$ be a connected graph. Let $e \in E$ be an edge of $G$. Then $e$ is a bridge {{iff}} $e$ does not lie on any circuit of $G$.


Characteristics of Traversable Graph

A finite loop-multigraph is traversable {{iff}} it is connected and no more than two vertices are odd. Any Eulerian trail which is not an Eulerian circuit must start and end at an odd vertex.


Number of Edges of Regular Graph

An $r$-regular graph of order $n$ is of size $\dfrac {n r} 2$.


Graph is Bipartite iff No Odd Cycles

Let $G$ be an undirected graph. Then $G$ is bipartite {{iff}} it has no odd cycles.


Ore's Theorem

Let $G = \struct {V, E}$ be a simple graph of order $n \ge 3$. Let $G$ be an Ore graph, that is: :For each pair of non-adjacent vertices $u, v \in V$: :: $(1): \quad \deg u + \deg v \ge n$ Then $G$ is Hamiltonian.


Path in Tree is Unique

Let $T$ be a graph. Then $T$ is a tree {{iff}} there is exactly one path between any two vertices.


Size of Tree is One Less than Order

Let $T$ be a connected simple graph of order $n$. Then $T$ is a tree {{Iff}} the size of $T$ is $n-1$.


Connected Subgraph of Tree is Tree

Let $T$ be a tree. Let $S$ be a subgraph of $T$ such that $S$ is connected. Then $S$ is also a tree.


Finite Tree has Leaf Nodes

Every non-edgeless finite tree has at least two leaf nodes.


Size of Cycle Graph equals Order

The size of a cycle graph equals its order.


Path Graph from Cycle Graph

Removing one edge from a cycle graph leaves a path graph.


Cycle in Balanced Signed Graph

Let $G$ be a balanced signed graph. Let $C$ be a cycle in $G$. Then $C$ has an even number of negative edges.


Graph with Even Vertices Partitions into Cycles

Let $G = \struct {V, E}$ be a graph whose vertices are all even. Then its edge set $E$ can be partitioned into cycles, no two of which share an edge. The converse also holds: a graph which can be partitioned into cycles must have all its vertices even.


Condition for Bipartite Graph to be Hamiltonian

Let $G = \struct {A \mid B, E}$ be a bipartite graph. Let $G$ be Hamiltonian. Then $\card A = \card B$. That is, there is the same number of vertices in $A$ as there are in $B$.


Complete Hamiltonian Bipartite Graph

Let $K_{m, n}$ be a complete bipartite graph. Then $K_{m, n}$ is Hamiltonian iff $m = n > 1$.


Condition for Complete Bipartite Graph to be Semi-Hamiltonian

Let $K_{m, n}$ be a complete bipartite graph. Then $K_{m, n}$ is semi-Hamiltonian {{iff}} either: :$m = n = 1$ or: :$m = n + 1$ (or $n = m + 1$).


Equivalent Definitions for Finite Tree

Let $T$ be a finite tree of order $n$. The following statements are equivalent: : $(1): \quad T$ is connected and has no circuits. : $(2): \quad T$ has $n-1$ edges and has no circuits. : $(3): \quad T$ is connected and has $n-1$ edges. : $(4): \quad T$ is connected, and the removal of any one edge renders $T$ disconnected. : $(5): \quad$ Any two vertices of $T$ are connected by exactly one path. : $(6): \quad T$ has no circuits, but adding one edge creates a cycle.


Number of Edges in Forest

Let $F = \struct {V, E}$ be a forest with $n$ nodes and $m$ components. Then $F$ contains $n - m$ edges.


Cayley's Formula

The number of distinct labeled trees with $n$ nodes is $n^{n - 2}$.


Bijection between Prüfer Sequences and Labeled Trees

There is a one-to-one correspondence between Prüfer sequences and labeled trees. That is, every labeled tree has a unique Prüfer sequence that defines it, and every Prüfer sequence defines just one labeled tree.


Tree has Center or Bicenter

Every tree has either: : $(1): \quad$ Exactly one center or: : $(2): \quad$ Exactly one bicenter, but never both. That is, every tree is either central or bicentral.


Prim's Algorithm produces Minimum Spanning Tree

Prim's Algorithm always produces a minimum spanning tree.


Number of Hamilton Cycles in Complete Graph

For all $n \ge 3$, the number of distinct Hamilton cycles in the complete graph $K_n$ is $\dfrac {\left({n-1}\right)!} 2$.


Reflexive Euclidean Relation is Equivalence

A relation is an equivalence {{iff}} it is either left-Euclidean or right-Euclidean, and also reflexive.


Reflexive Relation is Serial

Every reflexive relation is also a serial relation.


Complement of Complete Bipartite Graph

Let $K_{p, q}$ be a complete bipartite graph. The complement of $K_{p, q}$ consists of a disconnected graph with two components: :The complete graph $K_p$ :The complete graph $K_q$.


Complement of Strict Total Ordering

Let $\left({S, \prec}\right)$ be a relational structure such that $\prec$ is a strict total ordering. Then the complement of $\prec$ is a weak total ordering.


Mahler's Inequality

The geometric mean of the termwise sum of two finite sequences of positive real numbers is never less than the sum of their two separate geometric means: :$\displaystyle \prod_{k \mathop = 1}^n \paren {x_k + y_k}^{1/n} \ge \prod_{k \mathop = 1}^n x_k^{1/n} + \prod_{k \mathop = 1}^n y_k^{1/n}$ where $x_k, y_k > 0$ for all $k$.


Equivalence of Definitions of Transitive Closure (Relation Theory)/Intersection is Smallest

Let $\RR$ be a relation on a set $S$. Then the intersection of all transitive relations on $S$ that contain $\RR$ is the smallest transitive relation on $S$ that contains $\RR$.


Intersection of Transitive Relations is Transitive

The intersection of two transitive relations is also a transitive relation.


Intersection of Reflexive Relations is Reflexive

The intersection of two reflexive relations is also a reflexive relation.


Intersection of Symmetric Relations is Symmetric

The intersection of two symmetric relations is also a symmetric relation.


Union of Reflexive Relations is Reflexive

The union of two reflexive relations is also a reflexive relation.


Union of Symmetric Relations is Symmetric

The union of two symmetric relations is also a symmetric relation.


Union of Transitive Relations Not Always Transitive

The union of transitive relations is not necessarily itself transitive.


Set System Closed under Intersection is Commutative Semigroup

Let $\mathcal S$ be a system of sets. Let $\mathcal S$ be such that: :$\forall A, B \in \mathcal S: A \cap B \in \mathcal S$ Then $\left({\mathcal S, \cap}\right)$ is a commutative semigroup.


Set System Closed under Union is Commutative Semigroup

Let $\mathcal S$ be a system of sets. Let $\mathcal S$ be such that: :$\forall A, B \in \mathcal S: A \cup B \in \mathcal S$ Then $\struct {\mathcal S, \cup}$ is a commutative semigroup.


Set System Closed under Symmetric Difference is Abelian Group

Let $\SS$ be a system of sets. Let $\SS$ be such that: :$\forall A, B \in \SS: A * B \in \SS$ where $A * B$ denotes the symmetric difference between $A$ and $B$. Then $\struct {\SS, *}$ is an abelian group.


Ring of Sets is Commutative Ring

A ring of sets $\left({\mathcal R, *, \cap}\right)$ is a commutative ring whose zero is $\varnothing$.


Union as Symmetric Difference with Intersection

Let $A$ and $B$ be sets. Then: :$A \cup B = \left({A * B}\right) * \left({A \cap B}\right)$ where: : $A \cup B$ denotes set union : $A \cap B$ denotes set intersection : $A * B$ denotes set symmetric difference


Set Difference as Symmetric Difference with Intersection

:$S \setminus T = S * \paren {S \cap T}$ where: :$S \setminus T$ denotes set difference :$S * T$ denotes set symmetric difference :$S \cap T$ denotes set intersection.


Unit of System of Sets is Unique

The unit of a system of sets, if it exists, is unique. If $U$ is the unit of a system of sets $\mathcal S$, then $\forall A \in \mathcal S: A \subseteq U$.


Power Set is Algebra of Sets

Let $S$ be a set. Let $\mathcal P \left({S}\right)$ be the power set of $S$. Then $\mathcal P \left({S}\right)$ is an algebra of sets where $S$ is the unit.


Power Set is Closed under Intersection

Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Then: :$\forall A, B \in \powerset S: A \cap B \in \powerset S$


Power Set is Closed under Symmetric Difference

Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Then: :$\forall A, B \in \powerset S: A * B \in \powerset S$ where $A * B$ is the symmetric difference between $A$ and $B$.


Intersection of Rings of Sets

Let $\mathcal R_k$ be a ring of sets, where $k$ is an element of an arbitrary set of indices. Then their intersection $\displaystyle \mathcal R = \bigcap_k \mathcal R_k$ is itself a ring of sets.


Minimal Ring Generated by System of Sets

Let $\SS$ be a non-empty system of sets. Then there is a unique ring of sets $\map \RR \SS$ which: :$(1): \quad$ contains $\SS$ :$(2): \quad$ is contained by every ring of sets which also contains $\SS$. This ring of sets $\map \RR \SS$ is called the '''minimal ring generated by $\SS$'''.


Menelaus's Theorem

Let $ABC$ be a triangle. Let points $D, E, F$ lie on lines $BC, AC, AB$ respectively (produced if necessary). Then $D, E$ and $F$ are collinear {{iff}}: : $\dfrac {AF} {FB} \cdot \dfrac {BD} {DC} \cdot \dfrac {CE} {EA} = -1$ In the above, the line segments $AF, BD, EA$ are determined to have negative length if they lie outside the line segments $AB, BC, CA$.


Lines Joining Equal and Parallel Straight Lines are Parallel

The straight lines joining equal and parallel straight lines at their endpoints, in the same direction, are themselves equal and parallel. {{:Euclid:Proposition/I/33}}


Opposite Sides and Angles of Parallelogram are Equal

The opposite sides and angles of a parallelogram are equal to one another, and either of its diameters bisects its area. {{:Euclid:Proposition/I/34}}


Equal Sized Triangles on Same Base have Same Height

Triangles of equal area which are on the same base, and on the same side of it, are also in the same parallels. {{:Euclid:Proposition/I/39}}


Parallelogram on Same Base as Triangle has Twice its Area

A parallelogram on the same base as a triangle, and in the same parallels, has twice the area of the triangle. {{:Euclid:Proposition/I/41}}


Construction of Parallelogram equal to Triangle in Given Angle

A parallelogram can be constructed in a given angle the same size as any given triangle. {{:Euclid:Proposition/I/42}}


Construction of Parallelogram in Given Angle equal to Given Polygon

A parallelogram can be constructed in a given angle the same size as any given polygon. {{:Euclid:Proposition/I/45}}


Square of Sum

:$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$


Square of Sum less Square

:$\forall x, y \in \R: \paren {2x + y} y = \paren {x + y}^2 - x^2$


Square of Difference

:$\forall x, y \in \R: \paren {x - y}^2 = x^2 - 2 x y + y^2$


Square of Sum with Double

:$\forall a, b \in \R: \paren {a + 2 b}^2 = a^2 + 4 a b + 4 b^2$


Construction of Square equal to Given Polygon

A square can be constructed the same size as any given polygon. {{:Euclid:Proposition/II/14}}


Finding Center of Circle

For any given circle, it is possible to find its center. {{:Euclid:Proposition/III/1}}


Perpendicular Bisector of Chord Passes Through Center

The perpendicular bisector of any chord of any given circle must pass through the center of that circle. {{:Euclid:Proposition/III/1/Porism}}


Conditions for Diameter to be Perpendicular Bisector

If in a circle a diameter bisects a chord (which is itself not a diameter), then it cuts it at right angles, and if it cuts it at right angles then it bisects it. {{:Euclid:Proposition/III/3}}


Chords do not Bisect Each Other

If in a circle two chords (which are not diameters) cut one another, then they do not bisect one another. {{:Euclid:Proposition/III/4}}


Concentric Circles do not Intersect

If two circles are concentric, they share no points on their circumferences. Alternatively, this can be worded: :If two concentric circles share one point on their circumferences, then they share them all (that is, they are the same circle).


Baire Category Theorem

Let $M = \struct {A, d}$ be a complete metric space. Then $M = \struct {A, d}$ is also a Baire space.


Baire Characterisation Theorem

A real-valued function $f$ defined on a Banach space $X$ is a Baire-$1$ function {{iff}}: : for every non-empty closed subset $K$ of $X$, the restriction of $f$ to $K$ has a point of continuity relative to the topology of $K$.


Ramsey's Theorem

In any coloring of the edges of a sufficiently large complete graph, one will find monochromatic complete subgraphs. For 2 colors, Ramsey's theorem states that for any pair of positive integers $\tuple {r, s}$, there exists a least positive integer $\map R {r, s}$ such that for any complete graph on $\map R {r, s}$ vertices, whose edges are colored red or blue, there exists either a complete subgraph on $r$ vertices which is entirely red, or a complete subgraph on $s$ vertices which is entirely blue. More generally, for any given number of colors $c$, and any given integers $n_1, \ldots, n_c$, there is a number $\map R {n_1, \ldots, n_c}$ such that: :if the edges of a complete graph of order $\map R {n_1, \ldots, n_c}$ are colored with $c$ different colours, then for some $i$ between $1$ and $c$, it must contain a complete subgraph of order $n_i$ whose edges are all color $i$. This number $\map R {n_1, \ldots, n_c}$ is called the Ramsey number for $n_1, \ldots, n_c$. The special case above has $c = 2$ (and $n_1 = r$ and $n_2 = s$). Here $\map R {r, s}$ signifies an integer that depends on both $r$ and $s$. It is understood to represent the smallest integer for which the theorem holds.


Schur's Theorem (Ramsey Theory)

Let $r$ be a positive integer. Then there exists a positive integer $S$ such that: :for every partition of the integers $\set {1, \ldots, S}$ into $r$ parts, one of the parts contains integers $x$, $y$ and $z$ such that: ::$x + y = z$


Brahmagupta's Formula

The area of a cyclic quadrilateral with sides of lengths $a, b, c, d$ is: :$\sqrt {\paren {s - a} \paren {s - b} \paren {s - c} \paren {s - d} }$ where $s$ is the semiperimeter: :$s = \dfrac {a + b + c + d} 2$


Euler-Binet Formula

The Fibonacci numbers have a closed-form solution: :$F_n = \dfrac {\phi^n - \paren {1 - \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1 / \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1}^n \phi^{-n} } {\sqrt 5}$ where $\phi$ is the golden mean. Putting $\hat \phi = 1 - \phi = -\dfrac 1 \phi$ this can be written: :$F_n = \dfrac {\phi^n - \hat \phi^n} {\sqrt 5}$


Hölder's Inequality for Sums

Let $p, q \in \R_{>0}$ be strictly positive real numbers such that: :$\dfrac 1 p + \dfrac 1 q = 1$ Let: :$\mathbf x = \sequence {x_n} \in \ell^p$ :$\mathbf y = \sequence {y_n} \in \ell^q$ where $\ell^p$ denotes the $p$-sequence space. Let $\norm {\mathbf x}_p$ denote the $p$-norm of $\mathbf x$. Then $\mathbf x \mathbf y = \sequence {x_n y_n} \in \ell^1$, and: :$\norm {\mathbf x \mathbf y}_1 \le \norm {\mathbf x}_p \norm {\mathbf y}_q$


Binet Form

Let $m \in \R$. Define: {{begin-eqn}} {{eqn | l = \Delta       | r = \sqrt {m^2 + 4} }} {{eqn | l = \alpha       | r = \frac {m + \Delta} 2 }} {{eqn | l = \beta       | r = \frac {m - \Delta} 2 }} {{end-eqn}}


Binet-Cauchy Identity

:$\displaystyle \left({\sum_{i \mathop = 1}^n a_i c_i}\right) \left({\sum_{j \mathop = 1}^n b_j d_j}\right) = \left({\sum_{i \mathop = 1}^n a_i d_i}\right) \left({\sum_{j \mathop = 1}^n b_j c_j}\right) + \sum_{1 \mathop \le i \mathop < j \mathop \le n} \left({a_i b_j - a_j b_i}\right) \left({c_i d_j - c_j d_i}\right)$ where all of the $a, b, c, d$ are elements of a commutative ring. Thus the identity holds for $\Z, \Q, \R, \C$.


Nicomachus's Theorem

{{begin-eqn}} {{eqn | l = 1^3       | r = 1 }} {{eqn | l = 2^3       | r = 3 + 5 }} {{eqn | l = 3^3       | r = 7 + 9 + 11 }} {{eqn | l = 4^3       | r = 13 + 15 + 17 + 19 }} {{eqn | l = \vdots       | o =  }} {{end-eqn}} In general: :$\forall n \in \N_{>0}: n^3 = \left({n^2 - n + 1}\right) + \left({n^2 - n + 3}\right) + \ldots + \left({n^2 + n - 1}\right)$ In particular, the first term for $\left({n + 1}\right)^3$ is $2$ greater than the last term for $n^3$.


Conservation of Angular Momentum

Newton's Laws of Motion imply the conservation of angular momentum in systems of masses in which no external force is acting.


Determinant of Combinatorial Matrix

Let $C_n$ be the combinatorial matrix of order $n$ given by: :$C_n = \begin{bmatrix} x + y & y & \cdots & y \\ y & x + y & \cdots & y \\ \vdots & \vdots & \ddots & \vdots \\ y & y & \cdots & x + y \end{bmatrix}$ Then the determinant of $C_n$ is given by: :$\det \left({C_n}\right) = x^{n-1} \left ({x + n y}\right)$


Power Set is Closed under Union

Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Then: :$\forall A, B \in \powerset S: A \cup B \in \powerset S$


Matrix Product with Adjugate Matrix

Let $R$ be a commutative ring with unity. Let $\mathbf A \in R^{n \times n}$ be a square matrix of order $n$. Let $\adj {\mathbf A}$ be its adjugate matrix. Then: {{begin-eqn}} {{eqn | l = \mathbf A \cdot \adj {\mathbf A}       | r = \map \det {\mathbf A} \cdot \mathbf I_n }} {{eqn | l = \adj {\mathbf A} \cdot \mathbf A       | r = \map \det {\mathbf A} \cdot \mathbf I_n }} {{end-eqn}} where $\map \det {\mathbf A}$ is the determinant of $\mathbf A$.


Square of Ones Matrix

Let $\mathbf J = \sqbrk 1_n$ be a square ones matrix of order $n$. Then $\mathbf J^2 = n \mathbf J$. That is: :$\begin{bmatrix} 1 & 1 & \cdots & 1 \\ 1 & 1 & \cdots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \cdots & 1 \end{bmatrix}^2 = \begin{bmatrix} n & n & \cdots & n \\ n & n & \cdots & n \\ \vdots & \vdots & \ddots & \vdots \\ n & n & \cdots & n \end{bmatrix}$


Inverse of Combinatorial Matrix

Let $C_n$ be the combinatorial matrix of order $n$ given by: :$C_n = \begin{bmatrix} x + y & y & \cdots & y \\ y & x + y & \cdots & y \\ \vdots & \vdots & \ddots & \vdots \\ y & y & \cdots & x + y \end{bmatrix}$ Then its inverse $C_n^{-1} = \sqbrk b_n$ can be specified as: :$b_{i j} = \dfrac {-y + \delta_{i j} \paren {x + n y} } {x \paren {x + n y} }$ where $\delta_{i j}$ is the Kronecker delta.


Inverse of Vandermonde Matrix

Let $V_n$ be the Vandermonde matrix of order $n$ given by: :$V_n = \begin {bmatrix}   x_1 & x_2 & \cdots & x_n \\   x_1^2 & x_2^2 & \cdots & x_n^2 \\ \vdots & \vdots & \ddots & \vdots \\   x_1^n & x_2^n & \cdots & x_n^n \end{bmatrix}$ Then its inverse $V_n^{-1} = \sqbrk b_n$ can be specified as: :$b_{i j} = \begin{cases} \paren {-1}^{j - 1} \paren {\dfrac {\displaystyle \sum_{\substack {1 \mathop \le m_1 \mathop < \ldots \mathop < m_{n - j} \mathop \le n \\ m_1, \ldots, m_{n - j} \mathop \ne i} } x_{m_1} \cdots x_{m_{n - j} } } {x_i \displaystyle \prod_{\substack {1 \mathop \le m \mathop \le n \\ m \mathop \ne i} } \paren {x_m - x_i} } } & : 1 \le  j < n \\ \qquad \qquad \qquad \dfrac 1 {x_i \displaystyle \prod_{\substack {1 \mathop \le m \mathop \le n \\ m \mathop \ne i} } \paren {x_i - x_m} } & : j = n \end{cases}$


Floor Function is Idempotent

:$\floor {\floor x} = \floor x$


Relation between Floor and Ceiling

Let $x \in \R$ be a real number. Let $\left \lfloor {x}\right \rfloor$ be the floor of $x$, and $\left \lceil {x}\right \rceil$ be the ceiling of $x$. Then the following results apply:


Range of Values of Floor Function

=== Number less than Integer iff Floor less than Integer === {{:Number less than Integer iff Floor less than Integer}} === Number not less than Integer iff Floor not less than Integer === {{:Number not less than Integer iff Floor not less than Integer}} === Integer equals Floor iff between Number and One Less === {{:Integer equals Floor iff between Number and One Less}} === Integer equals Floor iff Number between Integer and One More === {{:Integer equals Floor iff Number between Integer and One More}}


Range of Values of Ceiling Function

=== Number greater than Integer iff Ceiling greater than Integer === {{:Number greater than Integer iff Ceiling greater than Integer}} === Number not greater than Integer iff Ceiling not greater than Integer === {{:Number not greater than Integer iff Ceiling not greater than Integer}} === Integer equals Ceiling iff between Number and One More === {{:Integer equals Ceiling iff between Number and One More}} === Integer equals Ceiling iff Number between Integer and One Less === {{:Integer equals Ceiling iff Number between Integer and One Less}}


Steiner Inellipse is Unique

For every triangle, there is one and only one Steiner inellipse.


Kronecker's Lemma

Let $\sequence {x_n}$ be an infinite sequence of real numbers such that: :$\displaystyle \sum_{n \mathop = 1}^\infty x_n = s$  exists and is finite. Then for $0 < b_1 \le b_2 \le b_3 \le \ldots$ and $b_n \to \infty$:  :$\displaystyle \lim_{n \mathop \to \infty} \frac 1 {b_n} \sum_{k \mathop = 1}^n b_k x_k = 0$


Number of Permutations

Let $S$ be a set of $n$ elements. Let $r \in \N: r \le n$. Then the number of $r$-permutations of $S$ is: :${}^r P_n = \dfrac {n!} {\paren {n - r}!}$ When $r = n$, this becomes: :${}^n P_n = \dfrac {n!} {\paren {n - n}!} = n!$ Using the falling factorial symbol, this can also be expressed: :${}^r P_n = n^{\underline r}$


Construction of Permutations

The ${}^n P_n$ permutations of $n$ objects can be generated algorithmically. By Number of Permutations, that number is given by: :${}^n P_n = n!$ where $n!$ denotes the factorial of $n$. This will be demonstrated to hold.


De Polignac's Formula

Let $n!$ be the factorial of $n$. Let $p$ be a prime number. Then $p^\mu$ is a divisor of $n!$, and $p^{\mu + 1}$ is not, where: :$\displaystyle \mu = \sum_{k \mathop > 0} \floor {\frac n {p^k} }$ where $\floor {\, \cdot \,}$ denotes the floor function.


Sum of Binomial Coefficients over Lower Index

:$\displaystyle \sum_{i \mathop = 0}^n \binom n i = 2^n$


Alternating Sum and Difference of Binomial Coefficients for Given n

:$\displaystyle \forall n \in \Z: \sum_{i \mathop = 0}^n \left({-1}\right)^i \binom n i = \delta_{n 0}$


Symmetry Rule for Binomial Coefficients

Let $n \in \Z_{>0}, k \in \Z$. Then: :$\dbinom n k = \dbinom n {n - k}$


Factors of Binomial Coefficient

For all $r \in \R, k \in \Z$: :$k \dbinom r k = r \dbinom {r - 1} {k - 1}$ where $\dbinom r k$ is a binomial coefficient. Hence: :$\dbinom r k = \dfrac r k \dbinom {r - 1} {k - 1}$ (if $k \ne 0$) and: :$\dfrac 1 r \dbinom r k = \dfrac 1 k \dbinom {r - 1} {k - 1}$ (if $k \ne 0$ and $r \ne 0$)


Sum of r+k Choose k up to n

:$\displaystyle \forall n \in \Z: n \ge 0: \sum_{k \mathop = 0}^n \binom {r + k} k = \binom {r + n + 1} n$


Negated Upper Index of Binomial Coefficient

:$\dbinom r k = \paren {-1}^k \dbinom {k - r - 1} k$


Alternating Sum and Difference of r Choose k up to n

:$\displaystyle \sum_{k \mathop \le n} \paren {-1}^k \binom r k = \paren {-1}^n \binom {r - 1} n$


Product of r Choose m with m Choose k

:$\dbinom r m \dbinom m k = \dbinom r k \dbinom {r - k} {m - k}$


Chu-Vandermonde Identity

:$\displaystyle \sum_k \binom r k \binom s {n - k} = \binom {r + s} n$


Lucas' Theorem

Let $p$ be a prime number. Let $n, k \in \Z_{\ge 0}$. Then: :$\dbinom n k \equiv \dbinom {\floor {n / p} } {\floor {k / p} } \dbinom {n \bmod p} {k \bmod p} \pmod p$ where: :$\dbinom n k$ denotes a binomial coefficient :$n \bmod p$ denotes the modulo operation :$\floor \cdot$ denotes the floor function.


Congruence Modulo Real Number is Equivalence Relation

For all $z \in \R$, congruence modulo $z$ is an equivalence relation.


Filter Basis Generates Filter

Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $\BB \subset \powerset S$. Then: :$\FF = \set {V \subseteq S: \exists U \in \BB: U \subseteq V}$ is a filter on $S$ {{iff}}: :$(1): \quad \forall V_1, V_2 \in \BB: \exists U \in \BB: U \subseteq V_1 \cap V_2$ :$(2): \quad \O \notin \BB, \BB \ne \O$ That is, $\BB$ is a filter basis of the filter $\FF$, which is generated by $\BB$.


Limit Point iff Superfilter Converges

Let $T = \left({S, \tau}\right)$ be a topological space. Let $\mathcal F$ be a filter on $S$. Let $x \in S$. Then $x$ is a limit point of $\mathcal F$ {{iff}} there exists a superfilter $\mathcal F'$ of $\mathcal F$ on $S$ which converges to $x$.


Zorn's Lemma

Let $\struct {X, \preceq}, X \ne \O$ be a non-empty ordered set such that every non-empty chain in $X$ has an upper bound in $X$. Then $X$ has at least one maximal element.


Ultrafilter Lemma

Let $S$ be a set. Every filter on $S$ is contained in an ultrafilter on $S$.


Generated Topology is a Topology

Let $X$ be a set. Let $\mathcal S \subseteq \mathcal P \left({X}\right)$, where $\mathcal P \left({X}\right)$ is the power set of $X$. Let $\mathcal T_\mathcal S$ be the generated topology for $\mathcal S$. Then $\mathcal T_\mathcal S$ is a topology on $X$.


Equivalent Definitions of Synthetic Basis

Let $X$ be a set. Then for a subset $\BB \subseteq \powerset X$ of the power set of $X$, the following are equivalent: :$(1): \quad$ $\displaystyle \forall A, B \in \BB: \exists \AA \subseteq \BB: A \cap B = \bigcup \AA$. :$(2): \quad$ $\displaystyle \forall A, B \in \BB: \forall x \in A \cap B: \exists W \in \BB: x \in W \subseteq A \cap B$.


Product Distributes over Modulo Operation

Let $x, y, z \in \R$ be real numbers. Let $x \bmod y$ denote the modulo operation. Then: :$z \left({x \bmod y}\right) = \left({z x}\right) \bmod \left({z y}\right)$


Congruence by Product of Moduli

Let $a, b, m \in \Z$. Let $a \equiv b \pmod m$ denote that $a$ is congruent to $b$ modulo $m$. Then $\forall n \in \Z, n \ne 0$: : $a \equiv b \pmod m \iff a n \equiv b n \pmod {m n}$


Definite Integral of Partial Derivative

Let $\map f {x, y}$ and $\map {\dfrac {\partial f} {\partial x} } {x, y}$ be continuous functions of $x$ and $y$ on $D = \closedint {x_1} {x_2} \times \closedint a b$. Then: :$\displaystyle \frac \d {\d x} \int_a^b \map f {x, y} \rd y = \int_a^b \map {\frac {\partial f} {\partial x} } {x, y} \rd y$ for $x \in \closedint {x_1} {x_2}$.


Law of Inverses (Modulo Arithmetic)

Let $m, n \in \Z$. Then: :$\exists n' \in \Z: n n' \equiv d \pmod m$ where $d = \gcd \set {m, n}$.


Common Factor Cancelling in Congruence

Let $a, b, x, y, m \in \Z$. Let: :$a x \equiv b y \pmod m$ and $a \equiv b \pmod m$ where $a \equiv b \pmod m$ denotes that $a$ is congruent modulo $m$ to $b$. Then: :$x \equiv y \pmod {m / d}$ where $d = \gcd \set {a, m}$.


Chinese Remainder Theorem

Let $a, b, r, s \in \Z$. Let $r \perp s$ (that is, let $r$ and $s$ be coprime). Then: :$a \equiv b \pmod {r s}$ {{iff}} $a \equiv b \pmod r$ and $a \equiv b \pmod s$ where $a \equiv b \pmod r$ denotes that $a$ is congruent modulo $r$ to $b$.


Sum of Floors not greater than Floor of Sum

Let $\left \lfloor {x} \right \rfloor$ be the floor function. Then: :$\left \lfloor {x} \right \rfloor + \left \lfloor {y} \right \rfloor \le \left \lfloor {x + y} \right \rfloor$ The equality holds: :$\left \lfloor {x} \right \rfloor + \left \lfloor {y} \right \rfloor = \left \lfloor {x + y} \right \rfloor$ {{iff}}: :$x \bmod 1 + y \bmod 1 < 1$ where $x \bmod 1$ denotes the modulo operation.


Sum of Ceilings not less than Ceiling of Sum

Let $\left \lceil {x} \right \rceil$ be the ceiling function. Then: :$\left \lceil {x} \right \rceil + \left \lceil {y} \right \rceil \ge \left \lceil {x + y} \right \rceil$ The equality holds: :$\left \lceil {x} \right \rceil + \left \lceil {y} \right \rceil = \left \lceil {x + y} \right \rceil$ {{iff}} either: :$x \in \Z$ or $y \in \Z$ or: :$x \bmod 1 + y \bmod 1 > 1$ where $x \bmod 1$ denotes the modulo operation.


Integer to Power of p-1 over 2 Modulo p

Let $a \in \Z$. Let $p$ be an odd prime. Let $b = a^{\frac {\paren {p - 1} } 2}$. Then one of the following cases holds: :$b \bmod p = 0$ which happens exactly when $a \equiv 0 \pmod p$, or: :$b \bmod p = 1$ or: :$b \bmod p = p - 1$ where: :$b \bmod p$ denotes the modulo operation :$x \equiv y \pmod p$ denotes that $x$ is congruent modulo $p$ to $y$.


Image Filter is Filter

Let $X, Y$ be sets. Let $\mathcal P \left({X}\right)$ and $\mathcal P \left({Y}\right)$ be the power sets of $X$ and $Y$ respectively. Let $f: X \to Y$ be a mapping. Let $\mathcal F \subset \mathcal P \left({X}\right)$ be a filter on $X$. Then the image filter of $\mathcal F$ with respect to $f$: :$f \left[{\mathcal F}\right] := \left\{ {U \subseteq Y: f^{-1} \left[{U}\right] \in \mathcal F}\right\}$ is a filter on $Y$.


Filter on Product Space Converges to Point iff Projections Converge to Projections of Point

Let $\family{X_i}_{i \mathop \in I}$ be an indexed family of non-empty topological spaces where $I$ is an arbitrary index set. Let $\displaystyle X := \prod_{i \mathop \in I} X_i$ be the corresponding product space. Let $\FF$ be a filter on $X$. Let $x \in X$. Let $\pr_i: X \to X_i$ denote the projection from $X$ onto $X_i$. Then $\FF$ converges to $x$ {{iff}}: :for all $i \in I$ the image filter $\map {\pr_i} \FF$ converges to $x_i := \map {\pr_i} x$.


Image of Ultrafilter is Ultrafilter

Let $X, Y$ be two sets, $f: X \to Y$ a mapping and $\mathcal F$ an ultrafilter on $X$. Then the image filter $f \left({\mathcal F}\right)$ is an ultrafilter on $Y$.


Schur's Inequality

Let $x, y, z \in \R_{\ge 0}$ be positive real numbers. Let $t \in \R, t > 0$ be a (strictly) positive real number. Then: :$x^t \paren {x - y} \paren {x - z} + y^t \paren {y - z} \paren {y - x} + z^t \paren {z - x} \paren {z - y} \ge 0$ The equality holds {{iff}} either: : $x = y = z$ : Two of them are equal and the other is zero. When $t$ is a positive even integer, the inequality holds for ''all'' real numbers $x, y, z$.


Wallis's Product

{{begin-eqn}} {{eqn | l = \prod_{n \mathop = 1}^\infty \frac {2 n} {2 n - 1} \cdot \frac {2 n} {2 n + 1}       | r = \frac 2 1 \cdot \frac 2 3 \cdot \frac 4 3 \cdot \frac 4 5 \cdot \frac 6 5 \cdot \frac 6 7 \cdot \frac 8 7 \cdot \frac 8 9 \cdots       | c =  }} {{eqn | r = \frac \pi 2       | c =  }} {{end-eqn}}


Number of Regions in Plane Defined by Given Number of Lines

The maximum number $L_n$ of regions in the plane that can be defined by $n$ straight lines in the plane is: :$L_n = \dfrac {n \paren {n + 1} } 2 + 1$ {{OEIS|A000124}}


Forward Difference of Falling Factorial

Let $f: \R \to \R$ be a real function. Let $\Delta$ denote the forward difference operator. Let $x^{\underline m}$ be the $m$th falling factorial of $x$ Then: :$\map \Delta {x^{\underline m} } = m x^{\underline {m - 1} }$


Chen's Theorem

Every sufficiently large even integer is the sum of either: :$(1): \quad$ two primes, or  :$(2): \quad$ a prime and a semiprime.


Binomial Coefficient involving Prime

Let $p$ be a prime number. Let $\dbinom n p$ be a binomial coefficient. Then: :$\dbinom n p \equiv \floor {\dfrac n p} \pmod p$ where: :$\floor {\dfrac n p}$ denotes the floor function.


Binomial Coefficient of Prime Minus One Modulo Prime

Let $p$ be a prime number. Then: :$0 \le k \le p - 1 \implies \dbinom {p - 1} k \equiv \left({-1}\right)^k \pmod p$ where $\dbinom {p - 1} k$ denotes a binomial coefficient.


Number of Primes is Infinite

The number of primes is infinite.


Quotient and Remainder to Number Base

Let $n \in \Z: n > 0$ be an integer. Let $n$ be expressed in base $b$: :$\displaystyle n = \sum_{j \mathop = 0}^m {r_j b^j}$ i.e. :$n = \left[{r_m r_{m-1} \ldots r_2 r_1 r_0}\right]_b$ Then: : $\displaystyle \left \lfloor {\frac n b} \right \rfloor = \left[{r_m r_{m-1} \ldots r_2 r_1}\right]_b$ : $n \bmod b = r_0$ where: : $\left \lfloor {.} \right \rfloor$ denotes the floor function; : $n \bmod b$ denotes the modulo operation.


Factorial Divisible by Prime Power

Let $n \in \Z: n \ge 1$. Let $p$ be a prime number. Let $n$ be expressed in base $p$ notation: :$\displaystyle n = \sum_{j \mathop = 0}^m r_j p^j$ where $0 \le r_j < p$. Let $n!$ be the factorial of $n$. Let $p^\mu$ be the largest power of $p$ which divides $n!$, that is: :$p^\mu \divides n!$ :$p^{\mu + 1} \nmid n!$ Then: :$\mu = \dfrac {n - \map {s_p} n} {p - 1}$ where $\map {s_p} n$ is the digit sum of $n$.


Wilson's Theorem/Corollary 2

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $p$ be a prime factor of $n!$ with multiplicity $\mu$. Let $n$ be expressed in a base $p$ representation as: {{begin-eqn}} {{eqn | l = n       | r = \sum_{j \mathop = 0}^m a_j p^j       | c = where $0 \le a_j < p$ }} {{eqn | r = a_0 + a_1 p + a_2 p^2 + \cdots + a_m p^m       | c = for some $m > 0$ }} {{end-eqn}} Then: :$\dfrac {n!} {p^\mu} \equiv \paren {-1}^\mu a_0! a_1! \dotsb a_m! \pmod p$


Negative Number is Congruent to Modulus minus Number

:$\forall m, n \in \Z: -m \equiv n - m \pmod n$ where $\bmod n$ denotes congruence modulo $n$.


Gaussian Integers form Subring of Complex Numbers

The ring of Gaussian integers: :$\struct {\Z \sqbrk i, +, \times}$ forms a subring of the set of complex numbers $\C$.


Gaussian Rationals form Subfield of Complex Numbers

The set of Gaussian rationals $\Q \left[{i}\right]$, under the operations of complex addition and complex multiplication, forms a subfield of the set of complex numbers $\C$.


Complex Conjugation is Automorphism

Consider the field of complex numbers $\C$. The operation of complex conjugation: :$\forall z \in \C: z \mapsto \overline z$ is a field automorphism.


Tschirnhaus Transformation

Let $P_n \left({x}\right) = 0$ be a polynomial equation of order $n$: :$a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 = 0$ Then the substitution: $y = x + \dfrac {a_{n-1}} {n a_n}$ converts $P_n$ into a depressed polynomial: :$b_n y^n + b_{n-1} y^{n-1} + \cdots + b_1 y + b_0 = 0$ where $b_{n-1} = 0$. Such a substitution is called a '''Tschirnhaus transformation'''.


Elementary Properties of Probability Measure

Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$. The probability measure $\Pr$ of $\EE$ has the following properties:


Elementary Properties of Event Space

Let $\mathcal E$ be an experiment with a probability space $\left({\Omega, \Sigma, \Pr}\right)$. The event space $\Sigma$ of $\mathcal E$ has the following properties:


Ring of Sets is Semiring of Sets

Let $\mathcal R$ be a ring of sets. Then $\mathcal R$ is also a semiring of sets.


Set of All Real Intervals is Semiring of Sets

Let $\mathbb S$ be the set of all real intervals. Then $\mathbb S$ is a semiring of sets, but is '''not''' a ring of sets.


Ring of Sets Generated by Semiring

Let $\mathcal S$ be a semiring of sets. Let $\mathcal R \left({\mathcal S}\right)$ be the minimal ring generated by $\mathcal S$. Let $\mathcal L$ be the system of sets $A$ with the finite expansions: :$\displaystyle A = \bigcup_{k \mathop = 1}^n A_k$ with respect to the sets $A_k \in \mathcal S$. Then $\mathcal L = \mathcal R \left({\mathcal S}\right)$.


Ring of Sets Closed under Finite Intersection

Let $\mathcal R$ be a ring of sets. Let $A_1, A_2, \ldots, A_n \in \mathcal R$. Then: :$\displaystyle \bigcap_{j \mathop = 1}^n A_j \in \mathcal R$


Sigma-Algebra is Delta-Algebra

A $\sigma$-algebra is also a $\delta$-algebra.


Addition Law of Probability

:$\map \Pr {A \cup B} = \map \Pr A + \map \Pr B - \map \Pr {A \cap B}$


Cardinality is Additive Function

Let $S$ be a finite set. Let $\mathcal P \left({S}\right)$ be the power set of $S$. The function $C: \mathcal P \left({S}\right) \to \R$, where $C$ is defined as the cardinality of a set, is an additive function.


Finite Union of Sets in Additive Function

Let $\mathcal A$ be an algebra of sets. Let $f: \mathcal A \to \overline {\R}$ be an additive function. Let $A_1, A_2, \ldots, A_n$ be any finite collection of pairwise disjoint elements of $\mathcal A$. Then: :$\displaystyle f \left({\bigcup_{i \mathop = 1}^n A_i}\right) = \sum_{i \mathop = 1}^n f \left({A_i}\right)$ That is, for any collection of pairwise disjoint elements of $\mathcal A$, $f$ of their union equals the sum of $f$ of the individual elements.


Measure is Finitely Additive Function

Let $\Sigma$ be a $\sigma$-algebra on a set $X$. Let $\mu: \Sigma \to \overline {\R}$ be a measure on $\Sigma$. Then $\mu$ is finitely additive.


Finite Union of Sets in Subadditive Function

Let $\mathcal A$ be an algebra of sets. Let $f: \mathcal A \to \overline {\R}$ be a subadditive function. Let $A_1, A_2, \ldots, A_n$ be any finite collection of elements of $\mathcal A$. Then: :$\displaystyle f \left({\bigcup_{i \mathop = 1}^n A_i}\right) \le \sum_{i \mathop = 1}^n f \left({A_i}\right)$ That is, for any finite collection of elements of $\mathcal A$, $f$ of their union is less than or equal to the sum of $f$ of the individual elements.


Additive Function on Empty Set is Zero

Let $\mathcal A$ be an algebra of sets. Let $f: \mathcal A \to \overline {\R}$ be an additive function on $\mathcal A$. Then $f \left({\varnothing}\right) = 0$.


Additive Function is Strongly Additive

Let $\SS$ be an algebra of sets. Let $f: \SS \to \overline \R$ be an additive function on $\SS$. Then $f$ is also strongly additive. That is: :$\forall A, B \in \SS: \map f {A \cup B} + \map f {A \cap B} = \map f A + \map f B$


Set Difference is Disjoint with Reverse

:$\paren {S \setminus T} \cap \paren {T \setminus S} = \O$


Measure is Monotone

Let $\struct {X, \Sigma, \mu}$ be a measure space. Then $\mu$ is monotone, that is: :$\forall E, F \in \Sigma: E \subseteq F \implies \map \mu E \le \map \mu F$


Countably Additive Function also Finitely Additive

Let $\mathcal A$ be a $\sigma$-algebra. Let $f: \mathcal A \to \overline \R$ be a function, where $\overline \R$ denotes the set of extended real numbers. Let $f$ be a countably additive function: :$\displaystyle f \left({\bigcup_{i \mathop \in \N} A_i}\right) = \sum_{i \mathop \in \N} f \left({A_i}\right)$ such that there exists at least one $A \in \mathcal A$ such that $f \left({A}\right)$ is a finite number. Then $f$ is a finitely additive function.


Countably Additive Function of Null Set

Let $\mathcal A$ be a $\sigma$-algebra. Let $f: \mathcal A \to \overline \R$ be a function, where $\overline \R$ denotes the extended set of real numbers. Let $f$ be a countably additive function: :$\displaystyle f \left({\bigcup_{i \mathop \in \N} A_i}\right) = \sum_{i \mathop \in \N} f \left({A_i}\right)$ such that there exists at least one $A \in \mathcal A$ where $f \left({A}\right)$ is a finite number. Then: : $f \left({\varnothing}\right) = 0$


Non-Negative Additive Function is Monotone

Let $\SS$ be an algebra of sets. Let $f: \SS \to \overline \R$ be an additive function, that is: :$\forall A, B \in \SS: A \cap B = \O \implies \map f {A \cup B} = \map f A + \map f B$ If $\forall A \in \SS: \map f A \ge 0$, then $f$ is monotone, that is: :$A \subseteq B \implies \map f A \le \map f B$


Power Set is Sigma-Algebra

The power set of a set is a sigma-algebra.


Probability Measure on Equiprobable Outcomes

Let $\struct {\Omega, \Sigma, \Pr}$ be an equiprobability space. Let $\card \Omega = n$. Then: :$\forall \omega \in \Omega: \map \Pr \omega = \dfrac 1 n$ :$\forall A \subseteq \Omega: \map \Pr A = \dfrac {\card A} n$.


Conditional Probability Defines Probability Space

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a measure space. Let $B \in \Sigma$ such that $\Pr \left({B}\right) > 0$. Let $Q: \Sigma \to \R$ be the real-valued function defined as: :$Q \left({A}\right) = \Pr \left({A \mid B}\right)$ where: :$\Pr \left({A \mid B}\right) = \dfrac {\Pr \left({A \cap B}\right)}{\Pr \left({B}\right)}$ is the conditional probability of $A$ given $B$. Then $\left({\Omega, \Sigma, Q}\right)$ is a probability space.


Bayes' Theorem

Let $\Pr$ be a probability measure on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $\map \Pr {A \mid B}$ denote the conditional probability of $A$ given $B$. Let $\map \Pr A > 0$ and $\map \Pr B > 0$. Then: :$\map \Pr {B \mid A} = \dfrac {\map \Pr {A \mid B} \, \map \Pr B} {\map \Pr A}$


Event Independence is Symmetric

Let $A$ and $B$ be events in a probability space. Let $A$ be independent of $B$. Then $B$ is independent of $A$. That is, '''is independent of''' is a symmetric relation.


Set Difference and Intersection form Partition

Let $S$ and $T$ be sets such that: :$S \setminus T \ne \O$ :$S \cap T \ne \O$ where $S \setminus T$ denotes set difference and $S \cap T$ denotes set intersection. Then $S \setminus T$ and $S \cap T$ form a partition of $S$.


Independent Events are Independent of Complement

Let $A$ and $B$ be events in a probability space $\struct {\Omega, \Sigma, \Pr}$. Then $A$ and $B$ are independent {{iff}} $A$ and $\Omega \setminus B$ are independent.


Probability of Independent Events Not Happening

Let $\EE = \struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $A_1, A_2, \ldots, A_m \in \Sigma$ be independent events in the event space of $\EE$. Then the probability of none of $A_1$ to $A_m$ occurring is: :$\displaystyle \prod_{i \mathop = 1}^m \paren {1 - \map \Pr {A_i} }$


Probability of Occurrence of At Least One Independent Event

Let $\EE = \struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $A_1, A_2, \ldots, A_m \in \Sigma$ be independent events in the event space of $\EE$. Then the probability of at least one of $A_1$ to $A_m$ occurring is: :$\displaystyle 1 - \prod_{i \mathop = 1}^m \paren {1 - \map \Pr {A_i} }$


Total Probability Theorem

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $\set {B_1, B_2, \ldots}$ be a partition of $\Omega$ such that $\forall i: \map \Pr {B_i} > 0$. Then: :$\displaystyle \forall A \in \Sigma: \map \Pr A = \sum_i \map \Pr {A \mid B_i} \, \map \Pr {B_i}$


Probability of Limit of Sequence of Events

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.


Boole's Inequality

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $A_1, A_2, \ldots, A_n$ be events in $\Sigma$. Then: :$\displaystyle \map \Pr {\bigcup_{i \mathop = 1}^n A_i} \le \sum_{i \mathop = 1}^n \map \Pr {A_i}$


Additive Nowhere Negative Function is Subadditive

Let $\AA$ be an algebra of sets. Let $f: \AA \to \overline {\R}$ be an additive function such that: :$\forall A \in \AA: \map f A \ge 0$ Then $f$ is subadditive.


Probability Measure is Subadditive

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Then $\Pr$ is a subadditive function.


Inclusion-Exclusion Principle

Let $\SS$ be an algebra of sets. Let $A_1, A_2, \ldots, A_n$ be finite sets. Let $f: \SS \to \R$ be an additive function. Then: {{begin-eqn}} {{eqn | l = \map f {\bigcup_{i \mathop = 1}^n A_i}       | r = \sum_{i \mathop = 1}^n \map f {A_i}       | c =  }} {{eqn | o =        | ro= -       | r = \sum_{1 \mathop \le i \mathop < j \mathop \le n} \map f {A_i \cap A_j}       | c =  }} {{eqn | o =        | ro= +       | r = \sum_{1 \mathop \le i \mathop < j \mathop < k \mathop \le n} \map f {A_i \cap A_j \cap A_k}       | c =  }} {{eqn | o =        | ro= \cdots       | c =  }} {{eqn | o =        | ro= +       | r = \paren {-1}^{n - 1} \map f {\bigcap_{i \mathop = 1}^n A_i}       | c =  }} {{end-eqn}}


Existence of Probability Space and Discrete Random Variable

Let $I$ be some indexing set. Let $S = \left\{{s_i: i \in I}\right\} \subset \R$ be a countable set of real numbers. Let $\left\{{\pi_i: i \in I}\right\} \subset \R$ be a countable set of real numbers which satisfies: :$\displaystyle \forall i \in I: \pi_i \ge 0, \sum_{i \mathop \in I} \pi_i = 1$ Then there exists a probability space $\left({\Omega, \Sigma, \Pr}\right)$ and a discrete random variable $X$ on $\left({\Omega, \Sigma, \Pr}\right)$ such that the probability mass function $p_X$ of $X$ is given by: {{begin-eqn}} {{eqn | l=p_X \left({s_i}\right)       | r=\pi_i       | c=if $i \in I$ }} {{eqn | l=p_X \left({s}\right)       | r=0       | c=if $s \notin S$ }} {{end-eqn}}


Sum and Product of Discrete Random Variables

Let $X$ and $Y$ be discrete random variables on the probability space $\left({\Omega, \Sigma, \Pr}\right)$.


Countable Function on Power Set of Sample Space is Discrete Random Variable

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space such that $\Sigma$ is the power set of $\Omega$. Let $f: \Omega \to \R$ be a function such that $\Img f$ is countable. Then $f$ is a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$.


Characteristic Function on Event is Discrete Random Variable

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. Let $E \in \Sigma$ be any event of $\left({\Omega, \Sigma, \Pr}\right)$. Let $\chi_E: \Omega \to \left\{{0, 1}\right\}$ be the characteristic function of $E$. Then $\chi_E$ is a discrete random variable on $\left({\Omega, \Sigma, \Pr}\right)$.


Poisson Distribution Gives Rise to Probability Mass Function

Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $X$ have the poisson distribution with parameter $\lambda$ (where $\lambda > 0$). Then $X$ gives rise to a probability mass function.


Geometric Distribution Gives Rise to Probability Mass Function

Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $X$ have the geometric distribution with parameter $p$ (where $0 < p < 1$). Then $X$ gives rise to a probability mass function.


Negative Binomial Distribution Gives Rise to Probability Mass Function/Second Form

Let $X$ have the negative binomial distribution (second form) with parameters $n$ and $p$ ($0 < p < 1$). Then $X$ gives rise to a probability mass function.


Bernoulli Process as Binomial Distribution

Let $\left \langle{X_i}\right \rangle$ be a finite Bernoulli process of length $n$ such that each of the $X_i$ in the sequence is a Bernoulli trial with parameter $p$. Then the number of successes in $\left \langle{X_i}\right \rangle$ is modelled by a binomial distribution with parameters $n$ and $p$. Hence it can be seen that: :$X \sim \operatorname{B} \left({1, p}\right)$ is the same thing as $X \sim \operatorname{Bern} \left({p}\right)$


Binomial Distribution Approximated by Poisson Distribution

Let $X$ be a discrete random variable which has the binomial distribution with parameters $n$ and $p$. Suppose $n$ is "very large" and $p$ is "very small", but $np$ of a "reasonable size". Then $X$ can be approximated by a Poisson distribution with parameter $\lambda$ where $\lambda = np$.


Bernoulli Process as Geometric Distribution

Let $\left \langle{X_i}\right \rangle$ be a Bernoulli process with parameter $p$. Let $\mathcal E$ be the experiment which consists of performing the Bernoulli trial $X_i$ until a failure occurs, and then stop. Let $k$ be the number of successes before a failure is encountered. Then $k$ is modelled by a geometric distribution with parameter $p$.


Function of Discrete Random Variable

Let $X$ be a discrete random variable on the probability space $\struct {\Omega, \Sigma, \Pr}$. Let $g: \R \to \R$ be any real function. Then $Y = g \sqbrk X$, defined as: :$\forall \omega \in \Omega: \map Y \omega = g \sqbrk {\map X \omega}$ is also a discrete random variable.


Expectation of Function of Discrete Random Variable

Let $X$ be a discrete random variable. Let $\expect X$ be the expectation of $X$. Let $g: \R \to \R$ be a real function. Then: :$\displaystyle \expect {g \sqbrk X} = \sum_{x \mathop \in \Omega_X} \map g x \, \map \Pr {X = x}$ whenever the sum is absolutely convergent.


Probability Mass Function of Function of Discrete Random Variable

Let $X$ be a discrete random variable. Let $Y = g \left({X}\right)$, where $g: \R \to \R$ is a real function. Then the probability mass function of $Y$ is given by: :$\displaystyle p_Y \left({y}\right) = \sum_{x \mathop \in g^{-1} \left({y}\right)} \Pr \left({X = x}\right)$


Variance as Expectation of Square minus Square of Expectation

Let $X$ be a random variable. Then the variance of $X$ can be expressed as: :$\var X = \expect {X^2} - \paren {\expect X}^2$ That is, it is the expectation of the square of $X$ minus the square of the expectation of $X$.


Derivative of Geometric Sequence

Let $x \in \R: \size x < 1$. Then: :$\displaystyle \sum_{n \mathop \ge 1} n x^{n - 1} = \frac 1 {\paren {1 - x}^2}$


Expectation of Shifted Geometric Distribution

Let $X$ be a discrete random variable with the shifted geometric distribution with parameter $p$. Then the expectation of $X$ is given by: :$\expect X = \dfrac 1 p$


Expectation of Bernoulli Distribution

Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$. Then the expectation of $X$ is given by: :$\expect X = p$


Expectation of Binomial Distribution

Let $X$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$ for some $n \in \N$ and $0 \le p \le 1$. Then the expectation of $X$ is given by: :$\expect X = n p$


Expectation of Poisson Distribution

Let $X$ be a discrete random variable with the Poisson distribution with parameter $\lambda$. Then the expectation of $X$ is given by: :$\expect X = \lambda$


Variance of Bernoulli Distribution

Let $X$ be a discrete random variable with the Bernoulli distribution with parameter $p$: :$X \sim \Bernoulli p$ Then the variance of $X$ is given by: :$\var X = p \paren {1 - p}$


Variance of Poisson Distribution

Let $X$ be a discrete random variable with the Poisson distribution with parameter $\lambda$. Then the variance of $X$ is given by: :$\var X = \lambda$


Variance of Binomial Distribution

Let $X$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$. Then the variance of $X$ is given by: :$\var X = n p \paren {1 - p}$


Sum of Expectations of Independent Trials

Let $\EE_1, \EE_2, \ldots, \EE_n$ be a sequence of experiments whose outcomes are independent of each other. Let $X_1, X_2, \ldots, X_n$ be discrete random variables on $\EE_1, \EE_2, \ldots, \EE_n$ respectively. Let $\expect {X_j}$ denote the expectation of $X_j$ for $j \in \set {1, 2, \ldots, n}$. Then we have, whenever both sides are defined: :$\displaystyle \expect {\sum_{j \mathop = 1}^n X_j} = \sum_{j \mathop = 1}^n \expect {X_j}$ That is, the sum of the expectations equals the expectation of the sum.


Sum of Variances of Independent Trials

Let $\EE_1, \EE_2, \ldots, \EE_n$ be a sequence of experiments whose outcomes are independent of each other. Let $X_1, X_2, \ldots, X_n$ be discrete random variables on $\EE_1, \EE_2, \ldots, \EE_n$ respectively. Let $\var {X_j}$ be the variance of $X_j$ for $j \in \set {1, 2, \ldots, n}$. Then: :$\displaystyle \var {\sum_{j \mathop = 1}^n X_j} = \sum_{j \mathop = 1}^n \var {X_j}$ That is, the sum of the variances equals the variance of the sum.


Variance of Shifted Geometric Distribution

Let $X$ be a discrete random variable with the shifted geometric distribution with parameter $p$. Then the variance of $X$ is given by: :$\var X = \dfrac {1 - p} {p^2}$


Expectation of Geometric Distribution

Let $X$ be a discrete random variable with the geometric distribution with parameter $p$. Then the expectation of $X$ is given by: :$E \left({X}\right) = \dfrac p {1-p}$


Variance of Geometric Distribution

Let $X$ be a discrete random variable with the geometric distribution with parameter $p$. Then the variance of $X$ is given by: :$\operatorname{var} \left({X}\right) = \dfrac p {\left({1 - p}\right)^2}$


Bernoulli Process as Negative Binomial Distribution

Let $\sequence {X_i}$ be a Bernoulli process with parameter $p$.


Properties of Cumulative Distribution Function

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$. Let $\map F X$ be the cumulative distribution function of $X$: :$\forall x \in \R: \map F x = \map \Pr {X \le x}$ Then the following conditions apply to $\map F X$:


Discrete Random Variable is Random Variable

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. Let $X$ be a discrete random variable on $\left({\Omega, \Sigma, \Pr}\right)$. Then $X$ fulfils the condition: :$\forall x \in \R: \left\{{\omega \in \Omega: X \left({\omega}\right) \le x}\right\} \in \Sigma$ That is, $X$ fulfils the condition for it to be a random variable.


Expectation of Discrete Uniform Distribution

Let $X$ be a discrete random variable with the discrete uniform distribution with parameter $n$. Then the expectation of $X$ is given by: :$\expect X = \dfrac {n + 1} 2$


Variance of Discrete Uniform Distribution

Let $X$ be a discrete random variable with the discrete uniform distribution with parameter $p$. Then the variance of $X$ is given by: :$\var X = \dfrac {n^2 - 1} {12}$


Probability Generating Function of Degenerate Distribution

Let $X$ be the degenerate distribution: :$\forall x \in \N: p_X \left({x}\right) = \begin{cases} 1 & : x = k \\ 0 & : x \ne k \end{cases}$ where $k \in \N$. Then the p.g.f. of $X$ is: :$\Pi_X \left({s}\right) = s^k$


Probability Generating Function of Bernoulli Distribution

Let $X$ be a discrete random variable with the Bernoulli distribution with parameter $p$. Then the p.g.f. of $X$ is: :$\Pi_X \left({s}\right) = q + ps$ where $q = 1 - p$.


Probability Generating Function of Binomial Distribution

Let $X$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$. Then the p.g.f. of $X$ is: :$\Pi_X \left({s}\right) = \left({q + ps}\right)^n$ where $q = 1 - p$.


Probability Generating Function of Shifted Geometric Distribution

Let $X$ be a discrete random variable with the shifted geometric distribution with parameter $p$. Then the p.g.f. of $X$ is: :$\map {\Pi_X} s = \dfrac {p s} {1 - q s}$ where $q = 1 - p$.


Probability Generating Function of Geometric Distribution

Let $X$ be a discrete random variable with the geometric distribution with parameter $p$. Then the p.g.f. of $X$ is: :$\map {\Pi_X} s = \dfrac q {1 - p s}$ where $q = 1 - p$.


Probability Generating Function of Poisson Distribution

Let $X$ be a discrete random variable with the Poisson distribution with parameter $\lambda$. Then the p.g.f. of $X$ is: :$\Pi_X \left({s}\right) = e^{-\lambda \left({1-s}\right)}$


Expectation of Discrete Random Variable from PGF

Let $X$ be a discrete random variable whose probability generating function is $\map {\Pi_X} s$. Then the expectation of $X$ is the value of the first derivative of $\map {\Pi_X} s$ {{WRT|Differentiation}} $s$ at $s = 1$. That is: :$\expect X = \map {\Pi'_X} 1$


Variance of Discrete Random Variable from PGF

Let $X$ be a discrete random variable whose probability generating function is $\Pi_X \left({s}\right)$. Then the variance of $X$ can be easily obtained from the value of the second derivative of $\Pi_X \left({s}\right)$ WRT $s$ at $x=1$: :$\operatorname{var} \left({X}\right) = \Pi''_X \left({1}\right) + \mu - \mu^2$ where $\mu = E \left({X}\right)$ is the expectation of $X$.


Properties of Probability Generating Function

Let $X$ be a discrete random variable whose probability generating function is $\Pi_X \left({s}\right)$. Then $\Pi_X \left({s}\right)$ has the following properties:


Nth Derivative of Mth Power

Let $m \in \Z$ be an integer such that $m \ge 0$. The $n$th derivative of $x^m$ {{WRT|Differentiation}} $x$ is: :$\dfrac {\d^n} {\d x^n} x^m = \begin{cases} m^{\underline n} x^{m - n} & : n \le m \\ 0 & : n > m \end{cases}$ where $m^{\underline n}$ denotes the falling factorial.


Derivative of Complex Power Series

Let $\xi \in \C$ be a complex number. Let $\sequence {a_n}$ be a sequence in $\C$. Let $\displaystyle f \paren z = \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n$ be a power series in a complex variable $z \in \C$ about $\xi$. Let $R$ be the radius of convergence of the series defining $f \paren z$. Let $\cmod {z - \xi} < R$. Then $f$ is complex-differentiable and its derivative is: :$\displaystyle f' \paren z = \sum_{n \mathop = 1}^\infty n a_n \paren {z - \xi}^{n - 1}$


Probability Generating Function of Discrete Uniform Distribution

Let $X$ be a discrete random variable with the discrete uniform distribution with parameter $n$. Then the p.g.f. of $X$ is: :$\displaystyle \map {\Pi_X} s = \frac {s \paren {1 - s^n} } {n \paren {1 - s} }$


Derivatives of PGF of Bernoulli Distribution

Let $X$ be a discrete random variable with the Bernoulli distribution with parameter $p$. Then the derivatives of the PGF of $X$ w.r.t. $s$ are: :$\dfrac {\mathrm d^k} {\mathrm d s^k} \Pi_X \left({s}\right) = \begin{cases} p & : k = 1 \\ 0 & : k > 1 \end{cases}$


Derivatives of PGF of Binomial Distribution

Let $X$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$. Then the derivatives of the PGF of $X$ w.r.t. $s$ are: :$\dfrac {\mathrm d^k} {\mathrm ds^k} \Pi_X \left({s}\right) = \begin{cases} n^{\underline k} p^k \left({q + ps}\right)^{n-k} & : k \le n \\ 0 & : k > n \end{cases}$ where: : $n^{\underline k}$ is the falling factorial : $q = 1 - p$


Higher Derivatives of Exponential Function

Let $\exp x$ be the exponential function. Then: :$\map {\dfrac {\d^n} {\d x^n} } {\exp x} = \exp x$


Derivatives of PGF of Poisson Distribution

Let $X$ be a discrete random variable with the Poisson distribution with parameter $\lambda$. Then the derivatives of the PGF of $X$ {{WRT|Differentiation}} $s$ are: :$\dfrac {d^k} {\d s^k} \, \map {\Pi_X} s = \lambda^k e^{- \lambda \paren {1 - s} }$


Derivatives of Function of a x + b

Let $f$ be a real function which is differentiable on $\R$. Let $a, b \in \R$ be constants. Then: : $\dfrac {\d^n} {\d x^n} \left({f \left({a x + b}\right)}\right) = a^n \dfrac {\d^n} {\d z^n} \left({f \left({z}\right)}\right)$ where $z = a x + b$.


Nth Derivative of Reciprocal of Mth Power

Let $m \in \Z$ be an integer such that $m > 0$. The $n$th derivative of $\dfrac 1 {x^m}$ {{WRT|Differentiation}} $x$ is: :$\dfrac {\d^n} {\d x^n} \dfrac 1 {x^m} = \dfrac {\paren {-1}^n m^{\overline n}} {x^{m + n}}$ where $m^{\overline n}$ denotes the rising factorial.


Derivatives of PGF of Geometric Distribution

Let $X$ be a discrete random variable with the geometric distribution with parameter $p$. Then the derivatives of the PGF of $X$ {{WRT|Differentiation}} $s$ are: :$\dfrac {\d^n} {\d s^n} \map {\Pi_X} s = \dfrac {q p^n n!} {\paren {1 - p s}^{n + 1} }$ where $q = 1 - p$.


Derivatives of PGF of Shifted Geometric Distribution

Let $X$ be a discrete random variable with the shifted geometric distribution with parameter $p$. Then the derivatives of the PGF of $X$ {{WRT|Differentiation}} $s$ are: :$\dfrac {\mathrm d^n} {\mathrm d s^n} \Pi_X \left({s}\right) = \dfrac {p q^{n - 1} \left({n - 1}\right)!} {\left({1 - q s}\right)^{n + 1} }$ where $q = 1 - p$.


Total Expectation Theorem

Let $\mathcal E = \struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $x$ be a discrete random variable on $\mathcal E$. Let $\set {B_1 \mid B_2 \mid \cdots}$ be a partition of $\omega$ such that $\map \Pr {B_i} > 0$ for each $i$. Then: :$\displaystyle \expect X = \sum_i \expect {X \mid B_i} \, \map \Pr {B_i}$ whenever this sum converges absolutely. In the above: :$\expect X$ denotes the expectation of $X$ :$\expect {X \mid B_i}$ denotes the conditional expectation of $X$ given $B_i$.