Definition:Right Angle/Perpendicular/Foot

The '''foot''' of a perpendicular is the point where it intersects the line to which it is at right angles. In the above diagram, the point $C$ is the '''foot''' of the perpendicular $CD$.


Definition:Right Angle/Perpendicular

{{EuclidDefinition|book = I|def = 10|name = Right Angle}} :400px In the above diagram, the line $CD$ has been constructed so as to be a '''perpendicular''' to the line $AB$.


Definition:Cosine/Definition from Triangle

:400px In the above right triangle, we are concerned about the angle $\theta$. The '''cosine''' of $\angle \theta$ is defined as being $\dfrac {\text{Adjacent}} {\text{Hypotenuse}}$.


{{:Definition:Cosine/Definition from Circle/First Quadrant}}


Definition:Cosine/Definition from Circle/First Quadrant

Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P = \tuple {x, y}$ be the point on $C$ in the first quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let $AP$ be the perpendicular from $P$ to the $y$-axis. Then the '''cosine''' of $\theta$ is defined as the length of $AP$.


Definition:Cosine/Definition from Circle/Second Quadrant

:500px Let $P = \tuple {x, y}$ be the point on $C$ in the second quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let $AP$ be the perpendicular from $P$ to the $y$-axis. Then the '''cosine''' of $\theta$ is defined as the length of $AP$.


Definition:Cosine/Definition from Circle/Third Quadrant

:500px Let $P = \tuple {x, y}$ be the point on $C$ in the third quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let $AP$ be the perpendicular from $P$ to the $y$-axis. Then the '''cosine''' of $\theta$ is defined as the length of $AP$.


Definition:Cosine/Definition from Circle/Fourth Quadrant

:500px Let $P = \tuple {x, y}$ be the point on $C$ in the fourth quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let $AP$ be the perpendicular from $P$ to the $y$-axis. Then the '''cosine''' of $\theta$ is defined as the length of $AP$.


Definition:Tangent Function/Definition from Triangle

:400px In the above right triangle, we are concerned about the angle $\theta$. The '''tangent''' of $\angle \theta$ is defined as being $\dfrac{\text{Opposite}} {\text{Adjacent}}$.


{{:Definition:Tangent Function/Definition from Circle/First Quadrant}}


Definition:Tangent Function/Definition from Circle/First Quadrant

Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P$ be the point on $C$ in the first quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \tuple {1, 0}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''tangent''' of $\theta$ is defined as the length of $AB$.


Definition:Tangent Function/Definition from Circle/Second Quadrant

Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P$ be the point on $C$ in the second quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \tuple {1, 0}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''tangent''' of $\theta$ is defined as the length of $AB$.


Definition:Tangent Function/Definition from Circle/Third Quadrant

Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P$ be the point on $C$ in the third quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \tuple {1, 0}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''tangent''' of $\theta$ is defined as the length of $AB$.


Definition:Tangent Function/Definition from Circle/Fourth Quadrant

Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P$ be the point on $C$ in the fourth quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \tuple {1, 0}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''tangent''' of $\theta$ is defined as the length of $AB$.


Definition:Cotangent/Definition from Triangle

:400px In the above right triangle, we are concerned about the angle $\theta$. The '''cotangent''' of $\angle \theta$ is defined as being $\dfrac {\text{Adjacent}} {\text{Opposite}}$.


{{:Definition:Cotangent/Definition from Circle/First Quadrant}}


Definition:Cotangent/Definition from Circle/First Quadrant

Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P$ be the point on $C$ in the first quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \tuple {0, 1}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''cotangent''' of $\theta$ is defined as the length of $AB$.


Definition:Cotangent/Definition from Circle/Second Quadrant

Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P$ be the point on $C$ in the second quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \tuple {0, 1}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''cotangent''' of $\theta$ is defined as the length of $AB$.


Definition:Cotangent/Definition from Circle/Third Quadrant

Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P$ be the point on $C$ in the third quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \tuple {0, 1}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''cotangent''' of $\theta$ is defined as the length of $AB$.


Definition:Cotangent/Definition from Circle/Fourth Quadrant

Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P$ be the point on $C$ in the fourth quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \tuple {0, 1}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''cotangent''' of $\theta$ is defined as the length of $AB$.


Definition:Secant Function/Definition from Triangle

:400px In the above right triangle, we are concerned about the angle $\theta$. The '''secant''' of $\angle \theta$ is defined as being $\dfrac{\text{Hypotenuse}} {\text{Adjacent}}$.


Definition:Cosecant/Definition from Triangle

:400px In the above right triangle, we are concerned about the angle $\theta$. The '''cosecant''' of $\angle \theta$ is defined as being $\dfrac {\text {Hypotenuse}} {\text {Opposite}}$.


{{:Definition:Secant Function/Definition from Circle/First Quadrant}}


{{:Definition:Cosecant/Definition from Circle/First Quadrant}}


Definition:Secant Function/Definition from Circle/First Quadrant

Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P$ be the point on $C$ in the first quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \tuple {1, 0}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''secant''' of $\theta$ is defined as the length of $OB$.


Definition:Secant Function/Definition from Circle/Second Quadrant

:500px Let $P$ be the point on $C$ in the second quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \tuple {1, 0}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''secant''' of $\theta$ is defined as the length of $OB$. As $OP$ needs to be produced in the opposite direction to $P$, the '''secant''' is therefore a negative number in the second quadrant.


Definition:Secant Function/Definition from Circle/Third Quadrant

:500px Let $P$ be the point on $C$ in the third quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \tuple {1, 0}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''secant''' of $\theta$ is defined as the length of $OB$. As $OP$ needs to be produced in the opposite direction to $P$, the '''secant''' is therefore a negative number in the third quadrant.


Definition:Secant Function/Definition from Circle/Fourth Quadrant

:500px Let $P$ be the point on $C$ in the fourth quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \tuple {1, 0}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''secant''' of $\theta$ is defined as the length of $OB$.


Definition:Cosecant/Definition from Circle/First Quadrant

Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P$ be the point on $C$ in the first quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \tuple {0, 1}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''cosecant''' of $\theta$ is defined as the length of $OB$.


Definition:Cosecant/Definition from Circle/Second Quadrant

:500px Let $P$ be the point on $C$ in the second quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \tuple {0, 1}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''cosecant''' of $\theta$ is defined as the length of $OB$.


Definition:Cosecant/Definition from Circle/Third Quadrant

:500px Let $P$ be the point on $C$ in the third quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \tuple {0, 1}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''secant''' of $\theta$ is defined as the length of $OB$. As $OP$ needs to be produced in the opposite direction to $P$, the '''cosecant''' is therefore a negative number in the third quadrant.


Definition:Cosecant/Definition from Circle/Fourth Quadrant

:500px Let $P$ be the point on $C$ in the fourth quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let a tangent line be drawn to touch $C$ at $A = \tuple {0, 1}$. Let $OP$ be produced to meet this tangent line at $B$. Then the '''secant''' of $\theta$ is defined as the length of $OB$. As $OP$ needs to be produced in the opposite direction to $P$, the '''cosecant''' is therefore a negative number in the fourth quadrant.


Definition:Inverse Secant/Real/Arcsecant

{{:Graph of Arcsecant Function}} From Shape of Secant Function, we have that $\sec x$ is continuous and strictly increasing on the intervals $\hointr 0 {\dfrac \pi 2}$ and $\hointl {\dfrac \pi 2} \pi$. From the same source, we also have that: :$\sec x \to + \infty$ as $x \to \dfrac \pi 2^-$ :$\sec x \to - \infty$ as $x \to \dfrac \pi 2^+$ Let $g: \hointr 0 {\dfrac \pi 2} \to \hointr 1 \to$ be the restriction of $\sec x$ to $\hointr 0 {\dfrac \pi 2}$. Let $h: \hointl {\dfrac \pi 2} \pi \to \hointl \gets {-1}$ be the restriction of $\sec x$ to $\hointl {\dfrac \pi 2} \pi$. Let $f: \closedint 0 \pi \setminus \dfrac \pi 2 \to \R \setminus \openint {-1} 1$: :$\map f x = \begin{cases} \map g x & : 0 \le x < \dfrac \pi 2 \\ \map h x & : \dfrac \pi 2 < x \le \pi \end{cases}$ From Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly increasing on $\hointr 1 \to$. From Inverse of Strictly Monotone Function, $\map h x$ admits an inverse function, which will be continuous and strictly increasing on $\hointl \gets {-1}$. As both the domain and range of $g$ and $h$ are disjoint, it follows that: :$\map {f^{-1} } x = \begin{cases} \map {g^{-1} } x & : x \ge 1 \\ \map {h^{-1} } x & : x \le -1 \end{cases}$ This function $f^{-1} \left({x}\right)$ is called '''arcsecant''' of $x$ and is written $\arcsec x$. Thus: :The domain of $\arcsec x$ is $\R \setminus \openint {-1} 1$ :The image of $\arcsec x$ is $\closedint 0 \pi \setminus \dfrac \pi 2$.


Definition:Inverse Cosecant/Real/Arccosecant

{{:Graph of Arccosecant Function}} From Shape of Cosecant Function, we have that $\csc x$ is continuous and strictly decreasing on the intervals $\hointr {-\dfrac \pi 2} 0$ and $\hointl 0 {\dfrac \pi 2}$. From the same source, we also have that: :$\csc x \to + \infty$ as $x \to 0^+$ :$\csc x \to - \infty$ as $x \to 0^-$ Let $g: \hointr {-\dfrac \pi 2} 0 \to \hointl {-\infty} {-1}$ be the restriction of $\csc x$ to $\hointr {-\dfrac \pi 2} 0$. Let $h: \hointl 0 {\dfrac \pi 2} \to \hointr 1 \infty$ be the restriction of $\csc x$ to $\hointl 0 {\dfrac \pi 2}$. Let $f: \closedint {-\dfrac \pi 2} {\dfrac \pi 2} \setminus \set 0 \to \R \setminus \openint {-1} 1$: :$\map f x = \begin{cases} \map g x & : -\dfrac \pi 2 \le x < 0 \\ \map h x & : 0 < x \le \dfrac \pi 2 \end{cases}$ From Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\hointl {-\infty} {-1}$. From Inverse of Strictly Monotone Function, $\map h x$ admits an inverse function, which will be continuous and strictly decreasing on $\hointr 1 \infty$. As both the domain and range of $g$ and $h$ are disjoint, it follows that: :$\map {f^{-1} } x = \begin{cases} \map {g^{-1} } x & : x \le -1 \\ \map {h^{-1} } x & : x \ge 1 \end{cases}$ This function $\map {f^{-1} } x$ is called '''arccosecant''' of $x$ and is written $\arccsc x$. Thus: :The domain of $\arccsc x$ is $\R \setminus \openint {-1} 1$ :The image of $\arccsc x$ is $\closedint {-\dfrac \pi 2} {\dfrac \pi 2} \setminus \set 0$.


Definition:Normal Set

Let $S$ be a set. Then $S$ is a '''normal set''' {{iff}}: :$S \notin S$ That is, $S$ is not an element of itself.


Definition:Abnormal Set

Let $S$ be a set. Then $S$ is an '''abnormal set''' {{iff}}: :$S \in S$ That is, $S$ is an element of itself.


Definition:Group of Rationals Modulo One

Define a relation $\sim$ on $\Q$ such that: :$\forall p, q \in \Q: p \sim q \iff p - q \in \Z$ Then $\left({\Q / \sim, +}\right)$ is a group referred to as the '''group of rationals modulo one'''.


Definition:Hyperbolic Function

There are six basic '''hyperbolic functions''', as follows: === Hyperbolic Sine === {{:Definition:Hyperbolic Sine}} === Hyperbolic Cosine === {{:Definition:Hyperbolic Cosine}} === Hyperbolic Tangent === {{:Definition:Hyperbolic Tangent}} === Hyperbolic Cotangent === {{:Definition:Hyperbolic Cotangent}} === Hyperbolic Secant === {{:Definition:Hyperbolic Secant}} === Hyperbolic Cosecant === {{:Definition:Hyperbolic Cosecant}}


Definition:Polar Coordinates/Polar Plane

A plane upon which a system of polar coordinates has been applied is known as a polar coordinate plane.


Definition:Azimuth

=== Azimuth (Polar Coordinates) === Another name for the angular coordinate in a system of polar coordinates: {{:Definition:Polar Coordinates/Angular Coordinate}} === Azimuth (Astronomy) === {{:Definition:Azimuth (Astronomy)}} Category:Definitions/Polar Coordinates Category:Definitions/Astronomy 176b5pou9jmsdkjhrgkuoa577igox13


Definition:Irreducible Representation

Let $G$ be a group. Let $V$ be a $G$-representation. Then $V$ is '''irreducible''' iff the only subrepresentations are $V$ and $0$. Category:Definitions/Representation Theory 33aw8fk8gee83a3bx6abf6l2qjq0y39


Definition:Half-Integer

A '''half-integer''' is a number of the form: :$n + \dfrac 1 2$ where $n \in \Z$. The set of all '''half-integers''' can be defined as: :$\left\{{n + \dfrac 1 2: n \in \Z}\right\}$ and can be denoted as $\Z + \dfrac 1 2$.


Definition:Muggle

A '''muggle''' is a person who is mathematically illiterate. Specifically, it refers to those who do not realise that they are indeed embarrassingly mathematically unsophisticated, and insist on peppering their speech with mathematical-sounding phrases, like ''Q.E.D'' (on believing they have won an argument) or ''I had $n$ pints of beer last night'' (being under the impression that $n$ means "a big unspecified number"). It may be argued that it is unfair and elitist to call such people with such a demeaning cognomen, but we stand by our declaration on the grounds that it is frequently mathematicians who are equally unfairly derided by many in common society as being socially inept, awkward and lacking in sophistication. Such displays of utter balderdash in the media are exemplified in such productions as ''{{WP|The_Big_Bang_Theory|The Big Bang Theory}}'', which, while attempting to make braininess accessible to the averagely intellectually endowed, end up propagating the very stereotype which it purports to debunk.


Definition:Subset/Superset

If $S$ is a subset of $T$, then $T$ is a '''superset''' of $S$. This can be expressed by the notation $T \supseteq S$. This can be interpreted as '''$T$ includes $S$''', or (more rarely) '''$T$ contains $S$'''. Thus $S \subseteq T$ and $T \supseteq S$ mean the same thing.


Definition:Proper Subset/Proper Superset

If $S$ is a proper subset of $T$, then $T$ is a '''proper superset''' of $S$. This can be expressed by the notation $T \supsetneqq S$. This can be interpreted as '''$T$ properly contains $S$'''.


Definition:Set Union/Family of Sets/Subsets of General Set

Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of subsets of a set $X$. Then the '''union''' of $\family {S_i}$ is defined as: :$\displaystyle \bigcup_{i \mathop \in I} S_i := \set {x \in X: \exists i \in I: x \in S_i}$


Definition:Set Union/Family of Sets/Universal Set

Let $\mathbb U$ be a universal set. Let $I$ be an indexing set. Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of subsets of $\mathbb U$. Then the '''union''' of $\family {S_i}$ is defined and denoted as: :$\displaystyle \bigcup_{i \mathop \in I} S_i := \set {x \in \mathbb U: \exists i \in I: x \in S_i}$


Definition:Set Intersection/Family of Sets/Universal Set

Let $\mathbb U$ be a universal set. Let $I$ be an indexing set. Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of subsets of $\mathbb U$. Then the '''intersection''' of $\family {S_i}$ is defined and denoted as: :$\displaystyle \bigcap_{i \mathop \in I} S_i := \set {x \in \mathbb U: \forall i \in I: x \in S_i}$


Definition:Set Intersection/Family of Sets/Subsets of General Set

Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of subsets of a set $X$. Then the '''intersection''' of $\family {S_i}$ is defined as: :$\displaystyle \bigcap_{i \mathop \in I} S_i := \set {x \in X: \exists i \in I: x \in S_i}$


Definition:Bijection/Definition 1

A mapping $f: S \to T$ is a '''bijection''' {{iff}} both: :$(1): \quad f$ is an injection and: :$(2): \quad f$ is a surjection.


Definition:Bijection/Definition 2

A mapping $f: S \to T$ is a '''bijection''' {{iff}}: :$f$ has both a left inverse and a right inverse.


Definition:Bijection/Definition 3

A mapping $f: S \to T$ is a '''bijection''' {{iff}}: :the inverse $f^{-1}$ of $f$ is a mapping from $T$ to $S$.


Definition:Bijection/Definition 5

A relation $f \subseteq S \times T$ is a '''bijection''' {{iff}}: :$(1): \quad$ for each $x \in S$ there exists one and only one $y \in T$ such that $\tuple {x, y} \in f$ :$(2): \quad$ for each $y \in T$ there exists one and only one $x \in S$ such that $\tuple {x, y} \in f$.


Definition:Bijection/Definition 4

A mapping $f \subseteq S \times T$ is a '''bijection''' {{iff}}: :for each $y \in T$ there exists one and only one $x \in S$ such that $\tuple {x, y} \in f$.


Definition:Subtraction/Natural Numbers

Let $\N$ be the set of natural numbers. Let $m, n \in \N$ such that $m \le n$. Let $p \in \N$ such that $n = m + p$. Then we define the operation '''subtraction''' as: :$n - m = p$ The natural number $p$ is known as the '''difference''' between $m$ and $n$.


Definition:Subtraction/Integers

The '''subtraction''' operation in the domain of integers $\Z$ is written "$-$". As the set of integers is the Inverse Completion of Natural Numbers, it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times \N$ where two tuples are equivalent if the difference between the two elements of each tuples is the same. Thus '''subtraction''' can be formally defined on $\Z$ as the operation induced on those equivalence classes as specified in the definition of integers. It follows that: :$\forall a, b, c, d \in \N: \eqclass {\tuple {a, b} } \boxminus - \eqclass {\tuple {c, d} } \boxminus = \eqclass {\tuple {a, b} } \boxminus + \tuple {-\eqclass {\tuple {c, d} } \boxminus} = \eqclass {\tuple {a, b} } \boxminus + \eqclass {\tuple {d, c} } \boxminus$ Thus '''integer subtraction''' is defined between all pairs of integers, such that: :$\forall x, y \in \Z: x - y = x + \paren {-y}$


Definition:Addition/Naturally Ordered Semigroup

Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup. The operation $\circ$ in $\left({S, \circ, \preceq}\right)$ is called '''addition'''.


Definition:Multiplication/Rational Numbers

The '''multiplication operation''' in the domain of rational numbers $\Q$ is written $\times$. Let $a = \dfrac p q, b = \dfrac r s$ where $p, q \in \Z, r, s \in \Z \setminus \set 0$. Then $a \times b$ is defined as $\dfrac p q \times \dfrac r s = \dfrac {p \times r} {q \times s}$. This definition follows from the definition of and proof of existence of the field of quotients of any integral domain, of which the set of integers is one.


Definition:Ring (Abstract Algebra)/Binding Priority

In order to simplify expressions involving both $+$ and $\circ$, it is the convention that ring product has a higher precedence than ring addition: :$a \circ b + c := \paren {a \circ b} + c$


Definition:Summation/Vacuous Summation

Take the summation: :$\displaystyle \sum _{\Phi \left({j}\right)} a_j$ where $\Phi \left({j}\right)$ is a propositional function of $j$. Suppose that there are no values of $j$ for which $\Phi \left({j}\right)$ is true. Then $\displaystyle \sum_{\Phi \left({j}\right)} a_j$ is defined as being $0$. This summation is called a '''vacuous summation'''. This is because: :$\forall a: a + 0 = a$ where $a$ is a number. Hence for all $j$ for which $\Phi \left({j}\right)$ is false, the sum is unaffected. This is most frequently seen in the form: :$\displaystyle \sum_{j \mathop = m}^n a_j = 0$ where $m > n$. In this case, $j$ can not at the same time be both greater than or equal to $m$ and less than or equal to $n$. Some sources consider such a treatment as abuse of notation.


Definition:Summation/Summand

The set of elements $\set {a_j \in S: 1 \le j \le n, \map R j}$ is called the '''summand'''.


Definition:Product Notation (Algebra)/Multiplicand

The set of elements $\left\{{a_j \in S: 1 \le j \le n, R \left({j}\right)}\right\}$ is called the '''multiplicand'''.


Definition:Product Notation (Algebra)/Vacuous Product

Take the composite expressed in product notation: :$\displaystyle \prod_{\map R j} a_j$ where $\map R j$ is a propositional function of $j$. Suppose that there are no values of $j$ for which $\map R j$ is true. Then $\displaystyle \prod_{\map R j} a_j$ is defined as being $1$. '''Beware:''' ''not'' zero. This composite is called a '''vacuous product'''. This is because: :$\forall a: a \times 1 = a$ where $a$ is a number. Hence for all $j$ for which $\map R j$ is false, the product is unaffected. This is most frequently seen in the form: :$\displaystyle \prod_{j \mathop = m}^n a_j = 1$ where $m > n$. In this case, $j$ can not at the same time be both greater than or equal to $m$ and less than or equal to $n$.


Definition:Associate/Integers

Let $x, y \in \Z$. Then '''$x$ is an associate of $y$''' {{iff}} they are both divisors of each other. That is, $x$ and $y$ are '''associates''' {{iff}} $x \divides y$ and $y \divides x$.


Definition:Composant/Continuum

$C$ is a '''composant''' of $H$ {{iff}}: :there exists some $p \in H$ such that $C$ contains all points $x \in S$ such that $x$ and $p$ are both contained in some proper subcontinua of $H$.


Definition:Composant/Point

Let $p \in H$. Then $C$ is the '''composant''' of $p$ {{iff}}: :$C$ is the union of all proper subcontinua of $H$ that contain $p$.


Definition:Subcontinuum/Proper Subcontinuum

Let $K$ be a subcontinuum of $H$ such that $K \ne H$. That is, let $K$ be a proper subset of $H$. Then $K$ is a '''proper subcontinuum''' of $H$.


Definition:Degenerate Continuum/Non-Degenerate

A '''non-degenerate continuum''' of $T$ is a continuum in $T$ containing more than one element.


Definition:Meager Space/Non-Meager

$A$ is '''non-meager in $T$''' {{iff}} it cannot be constructed as a countable union of subsets of $S$ which are nowhere dense in $T$. That is, $A$ is '''non-meager in $T$''' {{iff}} it is not meager in $T$.


Definition:Coprime/Integers/Relatively Composite

If $\gcd \left\{{a, b}\right\} > 1$, then $a$ and $b$ are '''relatively composite'''. That is, two integers are '''relatively composite''' if they are not coprime. {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book VII/14 - Relatively Composite}}'' {{EuclidDefRefNocat|VII|14|Relatively Composite}}


Definition:Prime Number/Definition 1

A '''prime number''' $p$ is a positive integer that has exactly two divisors which are themselves positive integers.


Definition:Prime Number/Definition 2

Let $p$ be a positive integer. Then $p$ is a prime number {{iff}} $p$ has exactly four integral divisors: $\pm 1$ and $\pm p$.


Definition:Prime Number/Definition 3

Let $p$ be a positive integer. Then $p$ is a prime number {{iff}}: : $\tau \left({p}\right) = 2$ where $\tau \left({p}\right)$ denotes the tau function of $p$.


Definition:Pairwise Coprime

=== GCD Domain === {{:Definition:Pairwise Coprime/GCD Domain}} === Euclidean Domain === {{:Definition:Pairwise Coprime/Euclidean Domain}} === Integers === {{:Definition:Pairwise Coprime/Integers}} Category:Definitions/Abstract Algebra Category:Definitions/Coprimality 0sonl2zoaxr0t5bd1szcb4dwe1udned


Definition:Pairwise Coprime/Integers

A set of integers $S$ is '''pairwise coprime''' {{iff}}: :$\forall x, y \in S: x \ne y \implies x \perp y$ where $x \perp y$ denotes that $x$ and $y$ are coprime.


Definition:Pairwise Coprime/GCD Domain

Let $\struct {D, +, \times}$ be a GCD domain. A subset $S \subseteq D$ is '''pairwise coprime (in $D$)''' {{iff}}: :$\forall x, y \in S: x \ne y \implies x \perp y$ where $x \perp y$ denotes that $x$ and $y$ are coprime.


Definition:Pairwise Coprime/Euclidean Domain

Let $\struct {D, +, \times}$ be a Euclidean domain. A subset $S \subseteq D$ is '''pairwise coprime (in $D$)''' {{iff}}: :$\forall x, y \in S: x \ne y \implies x \perp y$ where $x \perp y$ denotes that $x$ and $y$ are coprime.


Definition:Polynomial Congruence/Number of Solutions

Let $S = \left\{{b_1, b_2, \ldots, b_n}\right\}$ be a complete set of residues modulo $n$. The '''number of solutions''' of $P \left({x}\right) \equiv 0 \pmod n$ is the number of integers $b \in S$ for which $P \left({b}\right) \equiv 0 \pmod n$.


Definition:Signature (Logic)

Let $\mathcal L$ be a formal language. A choice of vocabulary for $\mathcal L$ is called a '''signature''' for $\mathcal L$. === Signature for Predicate Logic === {{:Definition:Signature (Logic)/Predicate Logic}}


Definition:Non-Archimedean/Norm (Vector Space)

A norm $\norm {\,\cdot\,} $ on a vector space $X$ is '''non-Archimedean''' {{iff}} $\norm {\, \cdot \,}$ satisfies the axiom: {{begin-axiom}} {{axiom | n = N4         | lc= Ultrametric Inequality:         | q = \forall x, y \in R         | ml= \norm {x + y}         | mo= \le         | mr= \max \set {\norm x, \norm y} }} {{end-axiom}}


Definition:Non-Archimedean/Norm (Vector Space)/Archimedean

A norm $\norm {\,\cdot\,} $ on a vector space $X$ is '''Archimedean''' {{iff}} it is not non-Archimedean.


Definition:P-adic Norm/P-adic Metric

The '''$p$-adic metric''' on $\Q$ is the metric induced by $\norm{\cdot}_p$: :$\forall x, y \in \Q: \map d {x, y} = \norm{x - y}_p$


Definition:Non-Archimedean/Metric/Archimedean

A metric is '''Archimedean''' {{iff}} it is not non-Archimedean.


Definition:Non-Archimedean/Metric

A metric $d$ on a metric space $X$ is '''non-Archimedean''' {{iff}}: :$\map d {x, y} \le \max \set {\map d {x, z}, \map d {y, z} }$ for all $x, y, z \in X$.


Definition:Max Operation/General Definition

Let $S^n$ be the cartesian $n$th power of $S$. The '''max operation''' is the $n$-ary operation on $\struct {S, \preceq}$ defined recursively as: :$\forall x := \family {x_i}_{1 \mathop \le i \mathop \le n} \in S^n: \map \max x = \begin{cases} x_1 & : n = 1 \\ \map \max {x_1, x_2} & : n = 2 \\ \map \max {\map \max {x_1, \ldots, x_{n - 1} }, x_n} & : n > 2 \\ \end{cases}$ where $\map \max {x, y}$ is the binary max operation on $S^2$.


Definition:Language of Propositional Logic/Alphabet/Letter

The letters of $\LL_0$, called '''propositional symbols''', can be any infinite collection $\PP_0$ of arbitrary symbols. It is usual to specify them as a limited subset of the English alphabet with appropriate subscripts. A typical set of '''propositional symbols''' would be, for example: :$\PP_0 = \set {p_1, p_2, p_3, \ldots, p_n, \ldots}$


Definition:Language of Propositional Logic/Alphabet/Sign/Bracket

{{begin-eqn}} {{eqn | ll= \bullet       | l = (       | o = :       | r = \)the '''left bracket''' sign\(       | c =  }} {{eqn | ll= \bullet       | l = )       | o = :       | r = \)the '''right bracket''' sign\(       | c =  }} {{end-eqn}}


Definition:Language of Propositional Logic/Alphabet/Sign/Connective

{{begin-eqn}} {{eqn | ll= \bullet       | l = \land       | o = :       | r = \)the conjunction sign\(       | c =  }} {{eqn | ll= \bullet       | l = \lor       | o = :       | r = \)the disjunction sign\(       | c =  }} {{eqn | ll= \bullet       | l = \implies       | o = :       | r = \)the conditional sign\(       | c =  }} {{eqn | ll= \bullet       | l = \iff       | o = :       | r = \)the biconditional sign\(       | c =  }} {{eqn | ll= \bullet       | l = \neg       | o = :       | r = \)the negation sign\(       | c =  }} {{eqn | ll= \bullet       | l = \top       | o = :       | r = \)the tautology sign\(       | c =  }} {{eqn | ll= \bullet       | l = \bot       | o = :       | r = \)the contradiction sign\(       | c =  }} {{end-eqn}} These comprise: * The nullary connectives $\top$ and $\bot$, representing the canonical tautology and contradiction, respectively * The unary connective $\neg$, representing negation * The binary connectives $\land, \lor, \implies$ and $\iff$, representing, respectively, conjunction, disjunction, implication and biconditional.


The alphabet $\mathcal A$ of the language of propositional logic $\mathcal L_0$ is defined as follows: <section notitle="true" name="tc"> === Letters === {{:Definition:Language of Propositional Logic/Alphabet/Letter}} {{transclude:Definition:Language of Propositional Logic/Alphabet/Sign   |section = def   |title = Signs   |header = 3   |link = true   |increase = 1 }} </section>


Definition:Language of Propositional Logic/Formal Grammar/Backus-Naur Form

In Backus-Naur form, the formal grammar of the language of propositional logic takes the following form: {{begin-eqn}} {{eqn | l = \)<tt><formula></tt>\(       | o = \)<tt>::=</tt>\(       | r = p \ \mid \ \top \ \mid \ \bot       | c = where $p \in \PP_0$ is a letter }} {{eqn | l = \)<tt><formula></tt>\(       | o = \)<tt>::=</tt>\(       | r = \neg\) <tt><formula></tt>\( }} {{eqn | l = \)<tt><formula></tt>\(       | o = \)<tt>::=</tt>\(       | r = (\) <tt><formula></tt> <tt><op></tt> <tt><formula></tt> \() }} {{eqn | l = \)<tt><op></tt>\(       | o = \)<tt>::=</tt>\(       | r = \land \ \mid \ \lor \ \mid \implies \mid \iff }} {{end-eqn}} Note that this is a top-down grammar: :we start with a metasymbol <tt><formula></tt> :progressively replace it with constructs containing other metasymbols and/or primitive symbols until finally we are left with a well-formed formula of $\LL_0$ consisting of nothing but primitive symbols.


Definition:Language of Propositional Logic/Formal Grammar/Bottom-Up Specification

The following rules of formation constitute a bottom-up grammar for the formation of well-formed formulas (WFFs) of the language of propositional logic $\LL_0$. :Let $\PP_0$ be the vocabulary of $\LL_0$. :Let $\mathrm {Op} = \set {\land, \lor, \implies, \iff}$. The rules are: {| |- | $\mathbf W: \text {TF}$ || | $:$ | $\top$ is a WFF, and $\bot$ is a WFF. |- | $\mathbf W: \PP_0$ || | $:$ | If $p \in \PP_0$, then $p$ is a WFF. |- | $\mathbf W: \neg$ || | $:$ | If $\mathbf A$ is a WFF, then $\neg \mathbf A$ is a WFF. |- | $\mathbf W: \text {Op}$ || | $:$ | If $\mathbf A$ is a WFF and $\mathbf B$ is a WFF and $\circ \in \mathrm {Op}$, then $\paren {\mathbf A \circ \mathbf B}$ is a WFF. |} Any string which can not be created by means of the above rules is not a WFF.


Definition:Language of Propositional Logic

In order to define $\mathcal L_0$, it is necessary to specify: * An alphabet $\mathcal A$ * A collation system with the unique readability property for $\mathcal A$ * A formal grammar (which determines the WFFs of $\mathcal L_0$)


Definition:Max Operation/General Definition/Real Numbers

The '''max operation''' on the real cartesian space $\R^n$ is the real-valued function $\max: \R^n \to \R$ defined recursively as: :$\forall x := \family {x_i}_{1 \mathop \le i \mathop \le n} \in \R^n: \map \max x = \begin{cases} x_1 & : n = 1 \\ \map \max {x_1, x_2} & : n = 2 \\ \map \max {\map \max {x_1, \ldots, x_{n - 1} }, x_n} & : n > 2 \\ \end{cases}$ where $\map \max {x, y}$ is the binary max operation on $\R \times \R$.


Definition:Cartesian Product/Cartesian Space/Real Cartesian Space

Let $n \in \N_{>0}$. Then $\R^n$ is the cartesian product defined as follows: :$\ds \R^n = \underbrace {\R \times \R \times \cdots \times \R}_{\text {$n$ times} } = \prod_{k \mathop = 1}^n \R$ Similarly, $\R^n$ can be defined as the set of all real $n$-tuples: :$\R^n = \set {\tuple {x_1, x_2, \ldots, x_n}: x_1, x_2, \ldots, x_n \in \R}$


Definition:Language of Propositional Logic/Keisler-Robbin

There are many formal languages expressing propositional logic. The formal language used on {{ProofWiki}} is defined on Definition:Language of Propositional Logic. This page defines the formal language $\LL_0$ used in: * {{BookReference|Mathematical Logic and Computability|1996|H. Jerome Keisler|author2 = Joel Robbin}} Explanations are omitted as this is intended for reference use only. === Alphabet === ==== Letters ==== The letters used are a non-empty set of symbols $\PP_0$. See the {{ProofWiki}} definition. ==== Signs ==== ===== Brackets ===== The '''brackets''' used are '''square brackets''': {{begin-eqn}} {{eqn | ll= \bullet       | l = [       | o = :       | r = \)the '''left bracket''' sign\(       | c =  }} {{eqn | ll= \bullet       | l = ]       | o = :       | r = \)the '''right bracket''' sign\(       | c =  }} {{end-eqn}} See the {{ProofWiki}} definition. ===== Connectives ===== The following '''connectives''' are used: {{begin-eqn}} {{eqn | ll= \bullet       | l = \land       | o = :       | r = \)the conjunction sign\(       | c =  }} {{eqn | ll= \bullet       | l = \lor       | o = :       | r = \)the disjunction sign\(       | c =  }} {{eqn | ll= \bullet       | l = \implies       | o = :       | r = \)the conditional sign\(       | c =  }} {{eqn | ll= \bullet       | l = \iff       | o = :       | r = \)the biconditional sign\(       | c =  }} {{eqn | ll= \bullet       | l = \neg       | o = :       | r = \)the negation sign\(       | c =  }} {{end-eqn}} See the {{ProofWiki}} definition. === Collation System === The collation system used is that of words and concatenation. See the {{ProofWiki}} definition. === Formal Grammar === The following bottom-up formal grammar is used. Let $\PP_0$ be the vocabulary of $\LL_0$. Let $Op = \set {\land, \lor, \implies, \iff}$. The rules are: {| |- | $\mathbf W: \PP_0$ || | $:$ | If $p \in \PP_0$, then $p$ is a WFF. |- | $\mathbf W: \neg$ || | $:$ | If $\mathbf A$ is a WFF, then $\neg \mathbf A$ is a WFF. |- | $\mathbf W: Op$ || | $:$ | If $\mathbf A$ and $\mathbf B$ are WFFs and $\circ \in Op$, then $\sqbrk {\mathbf A \circ \mathbf B}$ is a WFF. |} See the {{ProofWiki}} definition.


Definition:Order of Pole/Simple Pole

Let the order of the pole at $x$ be $1$. Then $x$ is a '''simple pole'''.


Definition:Order of Zero/Simple Zero

Let the order of the zero $x$ be $1$. Then $x$ is a '''simple zero'''.


Definition:Isolated Singularity/Pole

Let $z_0$ be an isolated singularity of $f$. Then $z_0$ is a '''pole''' {{iff}}: :$\displaystyle \lim_{z \mathop \to z_0} \cmod {\map f z} \to \infty$


Definition:Characteristic of Ring/Definition 1

Let $n \cdot x$ be defined as in Definition:Power of Element. The '''characteristic''' $\Char R$ of $R$ is the smallest $n \in \Z, n > 0$ such that $n \cdot 1_R = 0_R$. If there is no such $n$, then $\Char R = 0$.


Definition:Characteristic of Ring/Definition 2

Let $g: \Z \to R$ be the initial homomorphism, with $\map g n = n \cdot 1_R$. Let $\ideal p$ be the principal ideal of $\struct {\Z, +, \times}$ generated by $p$. The '''characteristic''' $\Char R$ of $R$ is the positive integer $p \in \Z_{\ge 0}$ such that $\ideal p$ is the kernel of $g$.


Definition:Characteristic of Ring/Definition 3

The '''characteristic of $R$''', denoted $\Char R$, is defined as follows. Let $p$ be the order of $1_R$ in the additive group $\struct {R, +}$ of $\struct {R, +, \circ}$. If $p \in \Z_{>0}$, then $\Char R := p$. If $1_R$ is of infinite order, then $\Char R := 0$.


Definition:Monoid Axioms

A monoid is an algebraic structure $\struct {S, \circ, e_S}$ which satisfies the following properties: {{begin-axiom}} {{axiom | n = \text S 0         | lc= Closure         | q = \forall a, b \in S         | m = a \circ b \in S }} {{axiom | n = \text S 1         | lc= Associativity         | q = \forall a, b, c \in S         | m = a \circ \paren {b \circ c} = \paren {a \circ b} \circ c }} {{axiom | n = \text S 2         | lc= Identity         | q = \exists e_S \in S: \forall a \in S         | m = e_S \circ a = a = a \circ e_S }} {{end-axiom}} The element $e_S$ is called the identity element.


Definition:Unique Parsability

Let $\mathcal L$ be a formal language. Let every WFF of $\mathcal L$ result from a unique rule of formation. Then $\mathcal L$ has '''unique parsability'''. === Bottom-Up Grammar === For a bottom-up grammar, a WFF $\phi$ results from a unique rule of formation iff: :$\phi$ results from applying the rule of formation $\mathbf R$ to WFFs $\phi_1, \ldots \phi_n$ :$\phi$ results from applying the rule of formation $\mathbf R'$ WFFs $\psi_1, \ldots \psi_m$ imply that $\mathbf R = \mathbf R'$, $m = n$, and for $i = 1, \ldots, n$, $\phi_i = \psi_i$. === Top-Down Grammar === For a top-down grammar, a WFF $\phi$ results from a unique rule of formation iff: :$\phi$ results from applying the rule of formation $\mathbf R$, substituting the WFFs $\phi_1, \ldots, \phi_n$ for the metasymbols of $\mathbf R$ :$\phi$ results from applying the rule of formation $\mathbf R'$, substituting the WFFs $\psi_1, \ldots, \psi_n$ for the metasymbols of $\mathbf R'$ imply that $\mathbf R = \mathbf R'$, $m = n$, and for $i = 1, \ldots, n$, $\phi_i = \psi_i$. Category:Definitions/Formal Systems qi4j9q92vrqo6xazzvorkgvractiw51


Definition:Bottom-Up Form of Top-Down Grammar

Let $\mathcal L$ be a formal language. Let $\mathcal L$ be given by a top-down formal grammar $\mathcal T$. The '''bottom-up form of $\mathcal T$''' is the formal grammar $\mathcal B$ defined by declaring that: :Each letter of $\mathcal L$ is a $\mathcal B$-WFF and, for each rule of formation $\mathbf R$ of $\mathcal T$ of the form: :A metasymbol may be replaced by the collation $\phi$ with metasymbols $\phi_1, \ldots, \phi_n$ declaring that $\mathcal B$ has the rule of formation $\mathbf R_{\mathcal B}$: :If $\phi_1, \ldots, \phi_n$ are $\mathcal B$-WFFs, so is $\phi$.


Definition:Semigroup Axioms

A semigroup is an algebraic structure $\struct {S, \circ}$ which satisfies the following properties: {{begin-axiom}} {{axiom | n = \text S 0         | lc= Closure         | q = \forall a, b \in S         | m = a \circ b \in S }} {{axiom | n = \text S 1         | lc= Associativity         | q = \forall a, b, c \in S         | m = a \circ \paren {b \circ c} = \paren  {a \circ b} \circ c }} {{end-axiom}}


Definition:Latin Square Property

Let $\left({S, \circ}\right)$ be an algebraic structure. $\left({S, \circ}\right)$ has the '''Latin square property''' {{iff}}: :$\forall a \in S$, the left and right regular representations $\lambda_a$ and $\rho_a$ are permutations on $S$. That is: :$\forall a, b \in S: \exists ! x: x \circ a = b$ :$\forall a, b \in S: \exists ! y: a \circ y = b$


Definition:Properties of Algebraic Structures of One Operation

The purpose of this page is to gather into one place the various types of algebraic structure of one binary operation and classify them according to the properties they hold. {| border="1" !  ! Closure ! Associativity ! Identity ! Inverses ! Commutativity ! Latin Square Property |- | Magma | align="center"| $\checkmark$ | | | | | |- | Quasigroup | align="center"| $\checkmark$ | | | | | align="center"| $\checkmark$ |- | Algebra Loop | align="center"| $\checkmark$ | | align="center"| $\checkmark$ | | | align="center"| $\checkmark$ |- | Semigroup | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | | | |- | Commutative Semigroup | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | | align="center"| $\checkmark$ |  |- | Monoid | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | | |- | Commutative Monoid | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | align="center"| $\checkmark$ | |- | Group | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | align="center"| $(\checkmark)$: see here |- | Abelian Group | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $(\checkmark)$: see here |- |} A '''checkmark''' in brackets: $(\checkmark)$ denotes that the property indicated can be derived from the others.


Definition:Properties of Algebraic Structures of Two Operations

The purpose of this page is to gather into one place the various types of algebraic structure of two binary operations: $+, *$, and to classify them according to the properties they hold. The properties are defined by the following key: {| border="1" | $A0$ | Closure of $+$ |- | $A1$ | Associativity of $+$ |- | $A2$ | Identity Element of $+$ |- | $A3$ | Inverses of $+$ |- | $A4$ | Commutativity of $+$ |- | $M0$ | Closure of $+$ |- | $M1$ | Associativity of $*$ |- | $M2$ | Identity Element of $*$ |- | $M3$ | Inverses of $*$ |- | $M4$ | Commutativity of $*$ |- | $D$ | Distributivity of $*$ over $+$ |- | $NZD$ | No (Proper) Zero Divisors |- | $AC2$ | Characteristic 2 for $+$ |- | $AZ$ | Zero Element of $+$ |- | $D'$ | Distributivity of $+$ over $*$ |- | $MI$ | Idempotence of $*$ |- | $MZ$ | Zero Element of $*$ |- |} {| border="1" !  ! $A0$ ! $A1$ ! $A2$ ! $A3$ ! $A4$ ! $M0$ ! $M1$ ! $M2$ ! $M3$ ! $M4$ ! $D$  ! $NZD$ ! $AC2$  ! $AZ$ ! $MI$ ! $MZ$ ! $D'$ |- | Semiring | align="center"| $\checkmark$ | align="center"| $\checkmark$ |  |  |  | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | |  | align="center"| $\checkmark$ |  |  | |  | |  |- | Additive Semiring | align="center"| $\checkmark$ | align="center"| $\checkmark$ |  |  | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | |  | align="center"| $\checkmark$ |  |  | |  | |  |- | Rig | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ |  | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | |  | align="center"| $\checkmark$ |  |  | |  | align="center"| $\checkmark$ |  |- | Ring without associativity | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ |  | | |  | align="center"| $\checkmark$ |  |  | |  | align="center"| $(\checkmark)$ (see here) |  |- | Ring | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | |  | align="center"| $\checkmark$ |  |  | |  | align="center"| $(\checkmark)$ (see here) |  |- | Ring with Unity | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | |  | align="center"| $\checkmark$ |  |  | |  | align="center"| $(\checkmark)$ (see here) |  |- | Commutative Ring with Unity | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | align="center"| $\checkmark$ | align="center"| $\checkmark$ |  |  | |  | align="center"| $(\checkmark)$ (see here) |  |- | Integral Domain | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ |  | |  | align="center"| $(\checkmark)$ (see here) |  |- | Division Ring | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | align="center"| $\checkmark$ | align="center"| $\checkmark$ |  | |  | align="center"| $(\checkmark)$ (see here) |  |- | Field | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ |  | |  | align="center"| $(\checkmark)$ (see here) |  |- | Idempotent Ring | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | |  |  | align="center"| $\checkmark$ | | align="center"| $\checkmark$ |  | align="center"| $\checkmark$ |  |  |- | Boolean Ring | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ |  |  | align="center"| $\checkmark$ | | align="center"| $\checkmark$ |  | align="center"| $\checkmark$ |  |  |- | Boolean Algebra | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ |- |} A '''checkmark''' in brackets: $(\checkmark)$ denotes that the property indicated can be derived from the others.


Definition:Support of Mapping to Algebraic Structure/Real-Valued Function

Let $S$ be a set. Let $f: S \to \R$ be a real-valued function. The '''support of $f$''' is the set of elements $x$ of $S$ whose values under $f$ are non-zero. That is: :$\operatorname{supp} \left({f}\right) := \left\{{x \in S: f \left({x}\right) \ne 0}\right\}$


Definition:Continuous Real-Valued Vector Function

Let $\R^n$ be the cartesian $n$-space. Let $f: \R^n \to \R$ be a real-valued function on $\R^n$. Then '''$f$ is continuous on $\R^n$''' iff: :$\forall a \in \R^n: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R^n: d \left({x, a}\right) < \delta \implies \left|{f \left({x}\right) - f \left({a}\right)}\right| < \epsilon$ where $d \left({x, a}\right)$ is the distance function on $\R^n$: :$\displaystyle d: \R^n \to \R: d \left({x, y}\right) := \sqrt {\left({\sum_{i \mathop = 1}^n \left({x_i - y_i}\right)}\right)}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right)$ are general elements of $\R^n$.


Definition:Support of Continuous Mapping/Real-Valued

Let $f: \R^n \to \R$ be a continuous real-valued function. The '''support of $f$''' is the closure of the set of elements $x$ of $\R^n$ whose values under $f$ are non-zero. That is: :$\operatorname{supp} \left({f}\right) = \overline{\left\{{x \in \R^n: f \left({x}\right) \ne 0}\right\}}$


Definition:Support of Distribution

Let $\Omega \subseteq \R^n$ be an open set. {{definition wanted|The link goes to a definition of an open set which has been made only for the real number line. Needs to be expanded to include the general real cartesian space. This has already been defined in the context of Metric Spaces; it's just a matter of creating a page which connects the concepts of an open set on a real number line to that on a general space.}} Let $\mathcal D \left({\Omega}\right)$ be the space of continuous functions compactly supported in $\Omega$. Let $T \in \mathcal D\,' \left({\Omega}\right)$ be a distribution. {{explain|$D\,'$: presumably a derivative?}} The '''support''' $\operatorname{supp} \left({T}\right) \subseteq \Omega$ of $T$ is defined by: : $\displaystyle x \notin \operatorname{supp} \left({T}\right)$ {{iff}}: :: there exists an open neighborhood $U$ of $x$ such that: ::: for all $\phi \in \mathcal D \left({\Omega}\right)$ such that $\operatorname{supp} \left({\phi}\right) \subseteq U$: :::: $T \left({\phi}\right) = 0$


Definition:Partition of Unity (Topology)/Subordinate

Let $X$ be a topological space. Let $\left\{{\phi_\alpha : \alpha \in A}\right\}$ be a partition of unity. Let $\mathcal B = \left\{{U_\beta: \beta \in B}\right\}$ be an open cover of $X$. Suppose the set $\left\{{\operatorname{supp} \left({\phi_\alpha}\right)^\circ : \alpha \in A}\right\}$ of interiors of supports is a refinement of $\mathcal B$ Then $\mathcal A$ is said to be '''subordinate''' to the cover $\mathcal B$.


Definition:Literal

A '''literal''' is either: :an atom $p$ of propositional logic, that is, a statement, or :the negation $\neg p$ of an atom $p$. In the language of propositional logic, these correspond to: :a letter $p$; :the WFF $\neg p$, where $p$ is a letter. === Positive Literal === {{:Definition:Literal/Positive}} === Negative Literal === {{:Definition:Literal/Negative}}


Definition:Conjunctive Normal Form

A propositional formula $P$ is in '''conjunctive normal form''' {{iff}} it consists of a conjunction of: :$(1):\quad$ disjunctions of literals and/or: :$(2):\quad$ literals.


Definition:Set of Truth Values

The '''set of truth values''' of propositional logic is the $2$-element set: :$\Bbb B = \set {\T, \F}$ of (Aristotelian) truth values.


Definition:Continued Fraction/Finite

Let $n \geq 0$ be a natural number. Informally, a '''finite continued fraction of length $n$''' in $F$ is an expression of the form: :$a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \cfrac 1 {\ddots \cfrac {} {a_{n-1} + \cfrac 1 {a_n}} }}}$ where $a_0, a_1, a_2, \ldots, a_n \in F$. Formally, a '''finite continued fraction of length $n$''' in $F$ is a finite sequence, called '''sequence of partial quotients''', whose domain is the integer interval $\left[0 \,.\,.\, n\right]$. A finite continued fraction should not be confused with its '''value''', when it exists.


Definition:Continued Fraction/Infinite

Informally, an '''infinite continued fraction''' in $F$ is an expression of the form: :$a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \cfrac 1 {\ddots \cfrac {} {a_{n-1} + \cfrac 1 {a_n + \cfrac 1 {\ddots}}} }}}$ where $a_0, a_1, a_2, \ldots, a_n, \ldots \in F$. Formally, an '''infinite continued fraction''' in $F$ is a sequence, called '''sequence of partial quotients''', whose domain is $\N_{\geq 0}$. An infinite continued fraction should not be confused with its '''value''', when it exists.


Definition:Partial Quotient

Let $(a_n)_{n\geq0}$ be a continued fraction in $k$, either finite or infinite. Let $n \geq 0$ be a natural number. The '''$n$th partial quotient''' is the $n$th term $a_n$.


Definition:Continued Fraction/Simple/Finite

Let $n\geq 0$ be a natural number. A '''simple finite continued fraction of length $n$''' is a finite continued fraction in $\R$ of length $n$ whose partial quotients are integers that are strictly positive, except perhaps the first. That is, it is a finite sequence $a : \left[0 \,.\,.\, n\right] \to \Z$ with $a_n > 0$ for $n >0$.


Definition:Continued Fraction/Simple/Infinite

A '''simple infinite continued fraction''' is a infinite continued fraction in $\R$ whose partial quotients are integers that are strictly positive, except perhaps the first. That is, it is a sequence $a : \N_{\geq 0} \to \Z$ with $a_n > 0$ for $n >0$.


Definition:Continued Fraction/Simple

Let $\R$ be the field of real numbers. === Simple Finite Continued Fraction === {{:Definition:Continued Fraction/Simple/Finite}} === Simple Infinite Continued Fraction === {{:Definition:Continued Fraction/Simple/Infinite}}


Definition:Value of Continued Fraction

Let $F$ be a field, such as the field of real numbers $\R$. === Finite Continued Fraction === {{Definition:Value of Continued Fraction/Finite}} === Infinite Continued Fraction === {{:Definition:Value of Continued Fraction/Infinite}}


Let $F$ be a field, such as the field of real numbers. Let $n \in \N \cup \{\infty\}$ be an extended natural number. Let $C = \left[{a_0, a_1, a_2, \ldots}\right]$ be a continued fraction in $F$ of length $n$. Let $k \leq n$ be a natural number. === Definition 1 === {{Definition:Convergent of Continued Fraction/Definition 1}} === Definition 2 === {{Definition:Convergent of Continued Fraction/Definition 2}}


Definition:Convergent of Continued Fraction/Even

The '''even convergents''' of $\left[{a_0, a_1, a_2, \ldots}\right]$ are the convergents $C_0, C_2, C_4, \ldots$, that is, those with an even subscript.


Definition:Convergent of Continued Fraction/Odd

The '''odd convergents''' of $\left[{a_0, a_1, a_2, \ldots}\right]$ are the convergents $C_1, C_3, C_5, \ldots$, that is, those with an odd subscript.


Definition:Numerators and Denominators of Continued Fraction

Let $F$ be a field. Let $n \in \N \cup \set \infty$ be an extended natural number. Let $C = \sqbrk {a_0, a_1, a_2, \ldots}$ be a continued fraction in $F$ of length $n$. === Definition 1: recursive definition === The '''sequence of numerators''' of $C$ is the sequence $\sequence {p_k}_{0 \mathop \le k \mathop \le n}$ that is recursively defined by: :$p_k = \begin {cases}   a_0 & : k = 0 \\   a_1 a_0 + 1 & : k = 1 \\   a_k p_{k - 1} + p_{k - 2} & : k \ge 2 \end {cases}$ The '''sequence of denominators''' of $C$ is the sequence $\sequence {q_k}_{0 \mathop \le k \mathop \le n}$ that is recursively defined by: :$q_k = \begin {cases}   1 & : k = 0 \\   a_1 & : k = 1 \\   a_k q_{k - 1} + q_{k - 2} & : k \ge 2 \end {cases}$ === Definition 2: using matrix products === Let $k \ge 0$, and let the indexed matrix product: :$\displaystyle \prod_{i \mathop = 0}^k \begin {pmatrix}   a_i & 1 \\   1 & 0 \end {pmatrix} = \begin {pmatrix}   x_{1 1}^{\paren k} & x_{1 2}^{\paren k} \\   x_{2 1}^{\paren k} & x_{2 2}^{\paren k} \end {pmatrix}$ The $k$th '''numerator''' is $x_{1 1}^{\paren k}$ and the $k$th '''denominator''' is $x_{2 1}^{\paren k}$.


Definition:Generalized Continued Fraction

Let $k$ be a field. Informally, a '''generalized continued fraction''' in $k$ is an expression of the form: :$b_0 + \cfrac {a_1} {b_1 + \cfrac {a_2} {b_2 + \cfrac {a_3} {\ddots \cfrac {} {b_{n-1} + \cfrac {a_n} {b_n + \cfrac {a_{n+1}} {\ddots}}} }}}$ Formally, a '''generalized continued fraction''' in $k$ is a pair of sequences $((b_n)_{n\geq 0}, (a_n)_{n\geq 1})$ in $k$, called '''sequence of partial denominators''' and '''sequence of partial numerators''' respectively. </onlyinclude>


Definition:Sheffer Operator

A '''Sheffer operator''' is a truth function which forms a functionally complete singleton set.


Definition:Functionally Complete

Let $S$ be a set of truth functions. Then $S$ is '''functionally complete''' {{iff}} all possible truth functions are definable from $S$.


Definition:Complement of Truth Function

Let $f: \Bbb B^k \to \Bbb B$ be a truth function. The '''complement''' of $f$ is the function $f'$ defined by: :$f': \Bbb B^k \to \Bbb B, f' \left({p}\right) = \neg \left({f \left({p}\right)}\right)$


=== Continuous Real-Valued Function in $\R^n$ === {{Definition:Support of Continuous Mapping/Real-Valued}} === General topological group === Let $X$ be a topological space. Let $G$ be a topological group with identity $e$. Let $f : X \to G$ be a continuous mapping. The '''support of $f$''' is the closure of the set of elements of $X$ that do not map to $e$ under $f$: :$\operatorname{supp} \left({f}\right) = \overline{\left\{{x \in X: f \left({x}\right) \ne e}\right\}}$


Definition:Propositional Function

A '''propositional function''' $\map P {x_1, x_2, \ldots}$ is an operation which acts on the objects denoted by the object variables $x_1, x_2, \ldots$ in a particular universe to return a truth value which depends on: :$(1): \quad$ The values of $x_1, x_2, \ldots$ :$(2): \quad$ The nature of $P$. === Satisfaction === {{:Definition:Propositional Function/Satisfaction}}


Definition:Boolean Interpretation/Formula

Let $v: \mathcal L_0 \to \left\{{T, F}\right\}$ be a (partial) boolean interpretation. Let $\phi$ be a WFF of propositional logic. Then $v$ is called a '''boolean interpretation for $\phi$''' iff $v$ is defined at $\phi$. Otherwise, $v$ is called a '''partial (boolean) interpretation for $\phi$'''.


Definition:Boolean Interpretation/Set of Formulas

Let $v: \mathcal L_0 \to \left\{{T, F}\right\}$ be a (partial) boolean interpretation. Let $\mathcal F$ be a set of WFFs of $\mathcal L_0$. Then $v$ is called a '''boolean interpretation for $\mathcal F$''' iff $v$ is defined on $\mathcal F$. Otherwise, $v$ is called a '''partial (boolean) interpretation for $\mathcal F$'''.


Definition:Boolean Interpretation/Truth Value

Let $\phi$ be a WFF of propositional logic. Let $v: \mathcal L_0 \to \left\{{T, F}\right\}$ be a boolean interpretation of $\phi$. The '''truth value of $\phi$ under $v$''' is $v (\phi)$.


Definition:Legendre Polynomial

The '''Legendre polynomials''' are the solutions to Legendre's differential equation.


=== Definition 1 === {{:Definition:Inverse Hyperbolic Sine/Complex/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Sine/Complex/Definition 2}}


Definition:Variable/Propositional Logic

A '''statement variable''' is a variable which is used to stand for arbitrary and unspecified statements. For '''statement variables''', lowercase letters are usually used, e.g.: :$p, q, r, \ldots{}$, etc. or lowercase Greek letters, e.g.: :$\phi, \psi, \chi$ etc. The citing of a variable can be interpreted as an assertion that the statement represented by that symbol is true.  That is: :'''$p$''' means :'''$p \text { is true}$'''


Definition:Scope (Logic)/Connective

Let $\mathcal L_0$ be the language of propositional logic. Let $\circ$ be a connective of $\mathcal L_0$. Let $\mathbf W$ be a well-formed formula of $\mathcal L_0$. The '''scope''' of an occurrence of $\circ$ in $\mathbf W$ is the smallest well-formed part of $\mathbf W$ containing this occurrence of $\circ$.


The WFFs of propositional logic can be made more readable by allowing them to be abbreviated. The resulting strings are not actually WFFs as such, but can be translated back uniquely into full WFFs. === Rules for Abbreviation of WFFs === {{:Definition:Abbreviation of WFFs of Propositional Logic/Rules}}


Definition:Abbreviation of WFFs of Propositional Logic/Standard Abbreviation

The string obtained by applying as many of the rules for abbreviation of WFFs  to a WFF $\mathbf A$ as possible is known as the '''standard abbreviation of $\mathbf A$'''.


Definition:Binding Priority/Propositional Logic

The binding priority convention which is almost universally used for the connectives of propositional logic is: :$(1): \quad \neg$ binds more tightly than $\lor$ and $\land$ :$(2): \quad \lor$ and $\land$ bind more tightly than $\implies$ and $\impliedby$ :$(3): \quad \implies$ and $\impliedby$ bind more tightly than $\iff$ Note that there is no overall convention defining which of $\land$ and $\lor$ bears a higher binding priority, and therefore we consider them to have equal priority. Because of this fact, ''unless specifically defined'', expressions such as $p \land q \lor r$ can not be interpreted unambiguously, and parenthesis ''must'' be used to determine the exact priorities which are to be used to interpret particular statements which may otherwise be ambiguous. Most sources do not recognise the use of $\impliedby$ as a separate connective from $\implies$, so the priority of one over the other is rarely a question.


Definition:Main Connective/Propositional Logic/Definition 1

Let $\mathbf C$ be a WFF of propositional logic. Let $\circ$ be a binary connective. Then $\circ$ is the '''main connective''' iff the scope of $\circ$ is $\mathbf C$.


Definition:Main Connective/Propositional Logic/Definition 2

Let $\mathbf C$ be a WFF of propositional logic such that: : $\mathbf C = \left({\mathbf A \circ \mathbf B}\right)$ where both $\mathbf A$ and $\mathbf B$ are both WFFs and $\circ$ is a binary connective. Then $\circ$ is the '''main connective''' of $\mathbf C$. Alternatively, let $\mathbf A$ be a WFF of propositional logic such that: : $\mathbf A = \neg \mathbf B$ where $\mathbf B$ is a WFF. Then $\neg$ is the '''main connective''' of $\mathbf A$.


Definition:Main Connective/Propositional Logic/Definition 3

Let $T$ be a WFF of propositional logic in the labeled tree specification. Suppose $T$ has more than one node. Then the label of the root of $T$ is called the '''main connective''' of $T$.


Definition:Classical Propositional Logic

'''Classical propositional logic''' is the branch of propositional logic based upon Aristotelian logic, whose philosophical tenets state that: :$(1): \quad$ A statement must be either true or false, and there can be no middle value. :$(2): \quad$ A statement cannot be both true and not true at the same time, that is, it may not contradict itself.


A '''labeled tree for propositional logic''' is a system containing: * A rooted tree $T$; * A countable set $\mathbf H$ of WFFs of propositional logic; * A WFF $\Phi \left({t}\right)$ attached to each non-root node $t$ of $T$. Such a structure can be denoted $\left({T, \mathbf H, \Phi}\right)$. === Hypothesis Set === {{:Definition:Labeled Tree for Propositional Logic/Hypothesis Set}} === Attached === {{:Definition:Labeled Tree for Propositional Logic/Attached}} === Child WFF === {{:Definition:Labeled Tree for Propositional Logic/Child WFF}} === Ancestor WFF === {{:Definition:Labeled Tree for Propositional Logic/Ancestor WFF}} === Along a Branch === {{:Definition:Labeled Tree for Propositional Logic/Along a Branch}}


Definition:Labeled Tree for Propositional Logic/Attached

Let $t$ be a non-root node of $T$. Let $\mathbf A$ be a WFF. Then '''$\mathbf A$ is attached to $t$''' {{iff}} $\mathbf A = \Phi \left({t}\right)$.


Definition:Labeled Tree for Propositional Logic/Child WFF

A WFF that is attached to a child of a node $t$ is called a '''child WFF of $t$'''.


Definition:Labeled Tree for Propositional Logic/Ancestor WFF

A WFF that is attached to an ancestor node of a node $t$ is called an '''ancestor WFF of $t$'''.


Definition:Labeled Tree for Propositional Logic/Along a Branch

Let $\Gamma$ be a branch of $T$. Let $\mathbf A$ be a WFF that is attached to a node $t \in \Gamma$. Then '''$\mathbf A$ occurs along $\Gamma$'''. This includes the case where $\mathbf A \in \mathbf H$, that is, $\mathbf A$ is attached to the root node.


Definition:Intuitionistic Propositional Logic

The '''intuitionist''' school of mathematics is one which adopts the following philosophical position: :"Although we may know that it is not the case that a statement $p$ is (provably) false, we don't necessarily know that it is (provably) true either." Thus the intuitionist school rejects the Law of the Excluded Middle. The classical school, by affirming that if a statement is not true it must be false, and if it is not false it must be true, accepts as an axiom that "not not-true" must mean "true". {{refactor|The below needs its own place (it's "Natural Deduction for Intuitionistic Propositional Logic")}} The system of '''intuitionistic propositional logic''' is, in consequence, based on the same axioms as that of classical propositional logic except for that disputed Law of the Excluded Middle.


Definition:Language of Propositional Logic/Labeled Tree

There are many formal languages expressing propositional logic. The formal language used on {{ProofWiki}} is defined on Definition:Language of Propositional Logic. This page defines the formal language $\LL_0$ used in: * {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed=3rd|edpage=Third Edition}} Explanations are omitted as this is intended for reference use only. === Alphabet === ==== Letters ==== The letters used are an infinite set of symbols $\PP_0$. This is the same as the {{ProofWiki}} definition. ==== Signs ==== ===== Connectives ===== The following '''connectives''' are used: {{begin-eqn}} {{eqn | ll= \bullet       | l = \neg       | o = :       | r = \)the negation sign\(       | c =  }} {{eqn | ll= \bullet       | l = \lor       | o = :       | r = \)the disjunction sign\(       | c =  }} {{eqn | ll= \bullet       | l = \land       | o = :       | r = \)the conjunction sign\(       | c =  }} {{eqn | ll= \bullet       | l = \implies       | o = :       | r = \)the conditional sign\(       | c =  }} {{eqn | ll= \bullet       | l = \iff       | o = :       | r = \)the biconditional sign\(       | c =  }} {{eqn | ll= \bullet       | l = \oplus       | o = :       | r = \)the exclusive or sign\(       | c =  }} {{eqn | ll= \bullet       | l = \downarrow       | o = :       | r = \)the nor sign\(       | c =  }} {{eqn | ll= \bullet       | l = \uparrow       | o = :       | r = \)the nand sign\(       | c =  }} {{end-eqn}} See the {{ProofWiki}} definition. === Collation System === The collation system used is that of labeled trees and adding ancestors. {{explain|"adding ancestors" signifies combining labeled trees into a new one by creating a common ancestor for their roots}} See the {{ProofWiki}} definition. === Formal Grammar === The following bottom-up formal grammar is used. Let $\PP_0$ be the vocabulary of $\LL_0$. The WFFs of $\LL_0$ are the smallest set $\FF$ of labeled trees such that: {{begin-axiom}} {{axiom | n = T1         | t = For each letter $p \in \PP_0$, the labeled tree with one node, whose label is $p$, is in $\FF$. }} {{axiom | n = T2         | t = For $T \in \FF$, the labeled tree obtained by adding an ancestor with label $\neg$ to the root node of $T$, is again in $\FF$. }} {{axiom | n = T3         | t = For $T_1 \in \FF$ and $T_2 \in \FF$ and a binary connective $\mathsf B$, the labeled tree obtained by adding a common ancestor labeled $\mathsf B$ of the root nodes of $T_1$ and $T_2$, is again in $\FF$. }} {{end-axiom}} Graphically, this means one has the following means to construct WFFs: ::$\begin{xy}\xymatrix{  p & &  \neg   \ar@{-}[d] & & &  \mathsf B   \ar@{-}[ld]   \ar@{-}[rd] \\ & &  <{\sf WFF}> & &   <{\sf WFF}> & &  <{\sf WFF}> }\end{xy}$ See the {{ProofWiki}} definition.


Definition:Absolute Value/Ordered Integral Domain

Let $\struct {D, +, \times, \le}$ be an ordered integral domain whose zero is $0_D$. Then for all $a \in D$, the '''absolute value''' of $a$ is defined as: :$\size a = \begin{cases} a & : 0_D \le a \\ -a & : a < 0_D \end{cases}$


Definition:Truth Function

Let $\mathbb B$ be the set of truth values, and let $k$ be a natural number. A mapping $f: \mathbb B^k \to \mathbb B$ is called a '''truth function'''.


Definition:Logical NAND/Truth Function

The NAND connective defines the truth function $f^\uparrow$ as follows: {{begin-eqn}} {{eqn | l=f^\uparrow \left({F, F}\right)       | r=T }} {{eqn | l=f^\uparrow \left({F, T}\right)       | r=T }} {{eqn | l=f^\uparrow \left({T, F}\right)       | r=T }} {{eqn | l=f^\uparrow \left({T, T}\right)       | r=F }} {{end-eqn}}


Definition:Logical NOR/Truth Function

The NOR connective defines the truth function $f^\downarrow$ as follows: {{begin-eqn}} {{eqn | l=f^\downarrow \left({F, F}\right)       | r=T }} {{eqn | l=f^\downarrow \left({F, T}\right)       | r=F }} {{eqn | l=f^\downarrow \left({T, F}\right)       | r=F }} {{eqn | l=f^\downarrow \left({T, T}\right)       | r=F }} {{end-eqn}}


Definition:Exclusive Or/Truth Function

The exclusive or connective defines the truth function $f^\oplus$ as follows: {{begin-eqn}} {{eqn | l=f^\oplus \left({F, F}\right)       | r=F }} {{eqn | l=f^\oplus \left({F, T}\right)       | r=T }} {{eqn | l=f^\oplus \left({T, F}\right)       | r=T }} {{eqn | l=f^\oplus \left({T, T}\right)       | r=F }} {{end-eqn}}


Definition:Truth Table/Characteristic

Let $\circledcirc$ be a logical connective. The '''characteristic truth table''' for $\circledcirc$ is the truth table describing the truth function of $\circledcirc$: :$\begin{array}{|cc||c|} \hline p & q & p \circledcirc q \\ \hline F & F & x \\ F & T & x \\ T & F & x \\ T & T & x \\ \hline \end{array}$ where $x$ is replaced by either $F$ or $T$ as appropriate on each row. The characteristic truth tables of the various logical connectives are listed below. === Logical Negation === {{:Definition:Logical Not/Truth Table}} === Logical Conjunction === {{:Definition:Conjunction/Truth Table}} === Logical Disjunction === {{:Definition:Disjunction/Truth Table}} === Biconditional === {{:Definition:Biconditional/Truth Table}} === Exclusive Disjunction === {{:Definition:Exclusive Or/Truth Table}} === Conditional === {{:Definition:Conditional/Truth Table}} === Logical NAND === {{:Definition:Logical NAND/Truth Table}} === Logical NOR === {{:Definition:Logical NOR/Truth Table}}


Definition:Generator of Monoid

Let $\left({M, \circ}\right)$ be a monoid Let $S \subseteq M$. Let $H$ be the smallest submonoid of $M$ such that $S \subseteq H$. Then: * $S$ is a '''generator'''  of $\left({H, \circ}\right)$ * $S$ '''generates''' $\left({H, \circ}\right)$ * $\left({H, \circ}\right)$ is the '''submonoid of $\left({M, \circ}\right)$ generated by $S$'''. This is written $H = \left\langle {S} \right\rangle$.


Definition:Boolean Interpretation

Let $\LL_0$ be the language of propositional logic, with vocabulary $\PP_0$. A '''boolean interpretation''' for $\LL_0$ is a propositional function: :$v: \PP_0 \to \set {\T, \F}$


Definition:Isomorphism (Abstract Algebra)/Monoid Isomorphism

Let $\struct {S, \circ}$ and $\struct {T, *}$ be monoids. Let $\phi: S \to T$ be a (monoid) homomorphism. Then $\phi$ is a monoid isomorphism {{iff}} $\phi$ is a bijection.


Definition:Monoid Automorphism

Let $\struct {S, \circ}$ be a monoid. Let $\phi: S \to S$ be a (monoid) isomorphism from $S$ to itself. Then $\phi$ is a monoid automorphism.


Definition:Additive Group of Real Numbers

The '''additive group of real numbers''' $\struct {\R, +}$ is the set of real numbers under the operation of addition.


Definition:Additive Group of Rational Numbers

The '''additive group of rational numbers''' $\left({\Q, +}\right)$ is the set of rational numbers under the operation of addition.


Definition:Multiplicative Group of Rational Numbers

The '''multiplicative group of rational numbers''' $\left({\Q_{\ne 0}, \times}\right)$ is the set of rational numbers without zero under the operation of multiplication.


Definition:Additive Group of Complex Numbers

The '''additive group of complex numbers''' $\struct {\C, +}$ is the set of complex numbers under the operation of addition.


Definition:Multiplicative Group of Complex Numbers

The '''multiplicative group of complex numbers''' $\struct {\C_{\ne 0}, \times}$ is the set of complex numbers without zero under the operation of multiplication.


Definition:Multiplicative Group of Real Numbers

The '''multiplicative group of real numbers''' $\struct {\R_{\ne 0}, \times}$ is the set of real numbers without zero under the operation of multiplication.


Definition:Field of Complex Numbers

The '''field of complex numbers''' $\struct {\C, +, \times}$ is the set of complex numbers under the two operations of addition and multiplication.


Definition:Field of Real Numbers

The '''field of real numbers''' $\struct {\R, + \times, \le}$ is the set of real numbers under the two operations of addition and multiplication, with an ordering $\le$ compatible with the ring structure of $\R$..


Definition:Field of Rational Numbers

The '''field of rational numbers''' $\struct {\Q, + \times, \le}$ is the set of rational numbers under the two operations of addition and multiplication, with an ordering $\le$ compatible with the ring structure of $\Q$.


Definition:Multiplicative Monoid of Integers Modulo m

Let $m \in \Z$ such that $m > 1$. The '''multiplicative monoid of integers modulo $m$''' $\struct {\Z_m, \times_m}$ is the set of integers modulo $m$ under the operation of multiplication modulo $m$.


Definition:Additive Group of Integers Modulo m

Let $m \in \Z$ such that $m > 1$. The '''additive group of integers modulo $m$''' $\struct {\Z_m, +_m}$ is the set of integers modulo $m$ under the operation of addition modulo $m$.


Definition:Radical Extension

Let $L / F$ be a field extension. Then $L$ is a '''radical extension of $F$''' iff there exist $\alpha_1, \ldots, \alpha_m \in F$ and $n_1, \ldots, n_2 \in \Z_{>0}$ such that: :$(1): \quad L = K \left[{\alpha_1, \ldots, \alpha_m}\right]$ :$(2): \quad \alpha_1^{n_1} \in F$ :$(3): \quad \forall i \in \N_m: \alpha_i^{n_i} \in F \left[{\alpha_1, \ldots, \alpha_{i-1}}\right]$ {{explain|link to definitions of $K \left[{\alpha_1, \ldots, \alpha_m}\right]$ and $F \left[{\alpha_1, \ldots, \alpha_{i-1} }\right]$}} Category:Definitions/Field Extensions mhhrn2hhw9k4rj8re7rflpi2nd0knsp


Definition:Additive Monoid of Natural Numbers

The '''additive monoid of natural numbers''' $\left({\N, +}\right)$ is the set of natural numbers under the operation of addition.


Definition:Language of Propositional Logic/Formal Grammar/WFF

Let $\mathbf A$ be approved of by the formal grammar of propositional logic. Then $\mathbf A$ is called a '''well-formed formula of propositional logic'''. Often, one abbreviates "well-formed formula", speaking of a '''WFF of propositional logic''' instead.


Definition:Statement Form/Substatement

A '''substatement''' of a statement form $\mathbf A$ is another statement form which occurs as a part of $\mathbf A$.


Definition:Finite Monoid

A '''finite monoid''' is a monoid whose underlying set is finite. Category:Definitions/Monoids 2fqcp6gr4u7mcakapzp30gtr2ap94ho


Definition:Additive Group of Integer Multiples

Let $n \in \Z_{>0}$. The '''additive group $\left({n \Z, +}\right)$ of integer multiples of $n$'''  is the set of integer multiples of $n$ under the operation of addition.


Definition:Root of Unity/Complex

Let $n \in \Z_{>0}$ be a (strictly) positive integer. The '''complex $n$th roots of unity''' are the elements of the set: :$U_n = \set {z \in \C: z^n = 1}$


Definition:Root of Unity/Primitive

A '''primitive $n$th root of unity of $F$''' is an element $\alpha \in U_n$ such that: :$U_n = \set {1, \alpha, \ldots, \alpha^{n - 1} }$


Definition:Affine Group of One Dimension

Let $S$ be the set of mappings $f_{a b}: \R \to \R$ defined as: :$S := \left\{{f_{a b}: x \mapsto a x + b: a \in \R_{\ne 0}, b \in \R}\right\}$ The algebraic structure $\left({S, \circ}\right)$, where $\circ$ denotes composition of mappings, is called the '''$1$-dimensional affine group on $\R$''' and can be denoted $\operatorname{Af}_1 \left({\R}\right)$.


Definition:Multiplicative Group of Positive Rational Numbers

The '''multiplicative group of positive rational numbers''' $\left({\Q_{> 0}, \times}\right)$ is the set of (strictly) positive rational numbers under the operation of multiplication.


Definition:Multiplicative Group of Positive Real Numbers

The '''multiplicative group of positive real numbers''' $\left({\R_{> 0}, \times}\right)$ is the set of (strictly) positive real numbers under the operation of multiplication.


Definition:Googol

A '''googol''' is defined to be $10^{100}$.


Definition:Googolplex

A '''googolplex''' is defined to be $10$ to the power of a googol, that is $10^{10^{100}}$ or $10^{\mathrm {googol}}$.


Definition:Hyperbola

=== Intersection with Cone === {{:Definition:Conic Section/Intersection with Cone}} {{:Definition:Conic Section/Intersection with Cone/Hyperbola}} === Focus-Directrix Property === {{:Definition:Conic Section/Focus-Directrix Property/Hyperbola}} === Equidistance Property === {{:Definition:Hyperbola/Equidistance}}


Definition:Parabola

=== Intersection with Cone === {{:Definition:Conic Section/Intersection with Cone}} {{:Definition:Conic Section/Intersection with Cone/Parabola}} === Focus-Directrix Property === {{:Definition:Conic Section/Focus-Directrix Property/Parabola}}


Definition:Centralizer/Group Subset

Let $\struct {G, \circ}$ be a group. Let $S \subseteq G$. The '''centralizer of $S$ (in $G$)''' is the set of elements of $G$ which commute with all $s \in S$: :$\map {C_G} S = \set {x \in G: \forall s \in S: x \circ s = s \circ x}$


Definition:Möbius Strip

A '''Möbius strip''' is a surface with boundary obtained by twisting a side of a rectangle by $180$ degrees, then joining the twisted side and the opposite side of it together. {{expand|An illustration of this process will aid understanding}}


Definition:Differentiable at Point

=== Real Function === {{:Definition:Differentiable Mapping/Real Function/Point}} === Complex Function === {{:Definition:Differentiable Mapping/Complex Function/Point}} === Real-Valued Function === {{:Definition:Differentiable Mapping/Real-Valued Function/Point}} === Between Manifolds === {{:Definition:Differentiable Mapping between Manifolds/Point}} Category:Definitions/Differential Calculus qkhwikxfxwgzmut1nsenhyu19q3cm2c


Definition:Faithful Group Action

Let $G$ be a group with identity $e$. Let $X$ be a set. Let $\phi: G \times X \to X$ be a group action. === Definition 1 === {{:Definition:Faithful Group Action/Definition 1}} === Definition 2 === {{:Definition:Faithful Group Action/Definition 2}}


Definition:Transitive Group Action

Let $G$ be a group. Let $S$ be a set. Let $*: G \times S \to S$ be a group action. The group action is '''transitive''' {{iff}} for any $x, y \in S$ there exists $g \in G$ such that $g * x = y$. That is, {{iff}} for all $x \in S$: :$\Orb x = S$ where $\Orb x$ denotes the orbit of $x \in S$ under $*$.


Definition:Trivial Group Action

Let $G$ be a group. Let $S$ be a set. Let $*: G \times S \to S$ be the group action defined as: :$\forall \tuple {g, s} \in G \times S: g * s = s$ Then $*$ is the '''trivial group action''' of $G$ on $S$.


Definition:Group Action Axioms

{{begin-axiom}} {{axiom | n = \text {GA} 1         | q = \forall g, h \in G, x \in X         | m = g * \paren {h * x} = \paren {g \circ h} * x }} {{axiom | n = \text {GA} 2         | q = \forall x \in X         | m = e * x = x }} {{end-axiom}}


Definition:Algebra of Sets/Definition 1

Let $X$ be a set. Let $\powerset X$ be the power set of $X$. Let $\RR \subseteq \powerset X$ be a set of subsets of $X$. Then $\RR$ is an '''algebra of sets over $X$''' {{iff}} the following conditions hold: {{begin-axiom}} {{axiom | n = \text {AS} 1         | lc= Unit:         | m = X \in \RR }} {{axiom | n = \text {AS} 2         | lc= Closure under Union:         | q = \forall A, B \in \RR         | m = A \cup B \in \RR }} {{axiom | n = \text {AS} 3         | lc= Closure under Complement Relative to $X$:         | q = \forall A \in \RR         | m = \relcomp X A \in \RR }} {{end-axiom}}


Definition:Algebra of Sets/Definition 2

An '''algebra of sets''' is a ring of sets with a unit.


Definition:Logical Argument/Finitary

A '''finitary argument''' is a logical argument which starts with a finite number of axioms, and can be translated into a finite number of statements.


Definition:Deductive Argument

A '''deductive argument''' is one whose conclusion follows unshakeable from the premises.


Definition:Argument Form

An '''argument form''' is a collation of symbols which contains statement variables such that: : when statements are used to replace statement variables (the same statement replacing the same statement variable throughout), the result is a logical argument. === Specific Form === {{:Definition:Argument Form/Specific Form}}


Definition:Argument Form/Specific Form

The '''specific form''' of a given logical argument is that argument form from which the logical argument results from replacing each distinct statement variable by a different simple statement.


Definition:Statement Form/Specific Form

The '''specific form''' of a given statement is that propositional formula from which the statement form results from replacing each distinct statement variable by a different simple statement.


Definition:Elementary Valid Argument

An '''elementary valid argument''' is a logical argument which is a substitution instance of an elementary valid argument form. The argument forms which are considered to be elementary valid argument forms will depend on the specific approach to propositional logic which is under discussion.


Definition:Operation/Arity/Ternary

A '''ternary operation''' (or '''three-place operation''') is an operation which takes three operands. That is, its arity is $3$.


Definition:Operation/Arity/Multiary

Certain types of operation have a variable number of operands. An operation which does not have a fixed arity is called '''multiary'''.


Definition:Operation/Arity/Finitary

A '''finitary operation''' is an operation which takes a finite number of operands.


Definition:Operation/Arity/Zero

It is possible to conceive of an operator which takes ''no'' operands. A constant can be considered as an operation which takes no operands. That is, it has an arity of zero.


Definition:Variable/Value

A variable $x$ may be (temporarily, conceptually) identified with a particular object. If so, then that object is called the '''value''' of $x$.


Definition:Propositional Function/Satisfaction

Let $\map P {x_1, x_2, \ldots, x_n}$ be an $n$-ary propositional function. If $a_1, a_2, \ldots, a_n$ have values which make $\map P {x_1, x_2, \ldots, x_n}$ true, then the ordered tuple $\tuple {a_1, a_2, \ldots, a_n}$ '''satisfies''' $\map P {x_1, x_2, \ldots, x_n}$.


Definition:Integer Division

Let $a, b \in \Z$ be integers such that $b \ne 0$.. From the Division Theorem: :$\exists_1 q, r \in \Z: a = q b + r, 0 \le r < \left|{b}\right|$ where $q$ is the quotient and $r$ is the remainder. The process of finding $q$ and $r$ is known as '''(integer) division'''.


Definition:Diagram (Graphical Technique)

A '''diagram''' is a graphical technique for illustrating a concept in picture form. It is generally considered that its use should be limited to that of an aid to understanding, and should not be used in order to prove something.


Definition:Quantified Statement

A '''quantified statement''' is a simple statement in predicate logic whose subject is qualified by either the universal quantifier or the existential quantifier. That is, it is either a universal statement or an existential statement. Category:Definitions/Predicate Logic 8uoinmnj34wnioqjbfik4kc9hgljjp5


Definition:Square of Opposition

The '''square of opposition''' is a diagram whose purpose is to illustrate the relations between the various categories of quantified statement. ::$\begin{xy} <-10em,0em>*+{\forall x: \map \Phi x} = "A", <10em,0em>*+{\forall x: \neg \map \Phi x} = "E", <-10em,-20em>*+{\exists x: \map \Phi x} = "I", <10em,-20em>*+{\exists x: \neg \map \Phi x} = "O", "A";"E" **@{-} ?<*@{<} ?>*@{>} ?*!/^.8em/{\text{Contraries}}, "A";"I" **@{-} ?>*@{>} ?*!/^3.2em/{\text{Subimplicant}}, "A";"O" **@{-} ?<*@{<} ?>*@{>} ?*!/^4em/{\text{Contradictories}}, "I";"E" **@{-} ?<*@{<} ?>*@{>} ?*!/_4em/{\text{Contradictories}}, "I";"O" **@{-} ?<*@{<} ?>*@{>} ?*!/_.8em/{\text{Subcontraries}}, "E";"O" **@{-} ?>*@{>} ?*!/^-3.2em/{\text{Subimplicant}}, \end{xy}$ This therefore illustrates the relations: :''All $x$ have $\Phi$'' is contrary to ''All $x$ do not have $\Phi$'' :''All $x$ have $\Phi$'' is contradictory to ''Some $x$ do not have $\Phi$'' :''Some $x$ have $\Phi$'' is contradictory to ''All $x$ do not have $\Phi$'' :''Some $x$ have $\Phi$'' is subimplicant to ''All $x$ have $\Phi$'' :''Some $x$ do not have $\Phi$'' is subimplicant to ''All $x$ do not have $\Phi$'' :''Some $x$ have $\Phi$'' is subcontrary to ''Some $x$ do not have $\Phi$'' where $x$ is an object and $\Phi$ is a property. === Vacuous Universe === ;Beware: Note that if the universe of discourse is empty, then the square of opposition no longer holds. Although $\forall x: \map \Phi x$ is vacuously true for such an empty universe, $\exists x: \map \Phi x$ is not. Thus $\exists x: \map \Phi x$ is no longer subimplicant to $\forall x: \map \Phi x$. Similarly, as $\exists x: \neg \map \Phi x$ is also false, it follows that $\exists x: \map \Phi x$ and $\exists x: \neg \map \Phi x$ are no longer subcontrary.


Definition:Predicate/Is of Identity

Consider the sentence: :'''Socrates is the philosopher who taught Plato.''' This could be reworded as: :'''The object named Socrates has the property of being the philosopher who taught Plato.''' However, the meaning that is ''really'' being conveyed here is that of: :'''The object named Socrates ''is the same object as'' the object which is the philosopher who taught Plato.''' In this context, '''is''' is not being used in the same way as the ''is'' of predication. When being used to indicate that one object is the same object as another object, '''is''' is called '''the ''is'' of identity'''. In this context, ''is'' means the same as equals.


Definition:Predicate/Is of Predication

Consider the statement: :'''Socrates is a man.''' This means: :'''The object named Socrates has the property of being a man.''' Thus we see that '''is''' here means '''has the property of being'''. In this context, '''is''' here is called '''the ''is'' of predication'''.


Definition:Universal Affirmative

:''Every $S$ is $P$'' where $S$ and $P$ are predicates.


Definition:Universal Affirmative/Set Theory

The universal affirmative $\forall x: \map S x \implies \map P x$ can be expressed in set language as: :$\set {x: \map S x} \subseteq \set {x: \map P x}$ or, more compactly: :$S \subseteq P$


Definition:Universal Negative

:''No $S$ is $P$'' where $S$ and $P$ are predicates.


Definition:Universal Negative/Set Theory

The universal negative $\forall x: \map S x \implies \neg \map P x$ can be expressed in set language as: :$\set {x: \map S x} \implies \set {x: \map P x} = \O$ or, more compactly: :$S \subseteq \map \complement P$


Definition:Particular Affirmative

:''Some $S$ is $P$'' where $S$ and $P$ are predicates.


Definition:Particular Affirmative/Set Theory

The particular affirmative $\exists x: \map S x \land \map P x$ can be expressed in set language as: :$\set {x: \map S x} \cap \set {x: \map P x} \ne \O$ or, more compactly: :$S \cap P \ne \O$


Definition:Particular Negative/Set Theory

The particular negative $\exists x: \map S x \land \neg \map P x$ can be expressed in set language as: :$\set {x: \map S x} \cap \set {x: \neg \map P x} \ne \O$ or, more compactly: :$S \cap \map \complement P \ne \O$


Definition:Particular Negative

:''Some $S$ is not $P$'' where $S$ and $P$ are predicates.


Definition:Categorical Syllogism

A '''categorical syllogism''' is a logical argument which is structured as follows: $(1): \quad$ It has exactly two premises and one conclusion. : The first premise is usually referred to as the major premise. : The second premise is usually referred to as the minor premise. $(2): \quad$ It concerns exactly three terms, which are usually denoted: {{begin-axiom}} {{axiom | q = P         | t = the primary term }} {{axiom | q = M         | t = the middle term }} {{axiom | q = S         | t = the secondary term }} {{end-axiom}} $(3): \quad$ Each of the premises and conclusion is a categorical statement.


Definition:Categorical Syllogism/Secondary Term

The '''secondary term''' of a categorical syllogism is the term that appears as the first predicate of the conclusion of the syllogism. It also appears once in one of the premises of the syllogism, traditionally the minor premise. It is usually denoted by $S$.


Definition:Categorical Syllogism/Middle Term

The '''middle term''' of a categorical syllogism is the term that does not appear in the conclusion of the syllogism. It appears once in each of the premises of the syllogism. It is usually denoted by $M$.


Definition:Categorical Syllogism/Primary Term

The '''primary term''' of a categorical syllogism is the term that appears as the second predicate of the conclusion of the syllogism. It also appears once in one of the premises of the syllogism, traditionally the major premise. It is usually denoted by $P$.


There are two premises in a categorical syllogism: === Major Premise === {{:Definition:Categorical Syllogism/Premises/Major Premise}} === Minor Premise === {{:Definition:Categorical Syllogism/Premises/Minor Premise}}


Definition:Categorical Syllogism/Terms

There are three terms in a categorical syllogism: === Primary Term of Syllogism === {{:Definition:Categorical Syllogism/Primary Term}} === Middle Term of Syllogism === {{:Definition:Categorical Syllogism/Middle Term}} === Secondary Term of Syllogism === {{:Definition:Categorical Syllogism/Secondary Term}}


Definition:Categorical Syllogism/Premises/Major Premise

The '''major premise''' of a categorical syllogism is conventionally stated first. It is a categorical statement which expresses the logical relationship between the primary term and the middle term of the syllogism.


Definition:Categorical Syllogism/Premises/Minor Premise

The '''minor premise''' of a categorical syllogism is conventionally stated second. It is a categorical statement which expresses the logical relationship between the secondary term and the middle term of the syllogism.


Definition:Categorical Syllogism/Conclusion

The '''conclusion''' of a categorical syllogism is stated last, conventionally prefaced with the word ''Therefore''. It is a categorical statement which expresses the logical relationship between the primary term and the secondary term of the syllogism. Furthermore, this categorical statement is ''specifically'' of the form in which the secondary term occurs ''first'' and the primary term occurs ''second''.


There are four patterns of categorical syllogism depending on the order of the terms in the major premise and minor premise. Let $\mathbf \Phi_1, \mathbf \Phi_2, \mathbf \Phi_3$ each be one of the categorical statements $\mathbf A$, $\mathbf E$, $\mathbf I$ or $\mathbf O$. Let $P$ denote the primary term, $S$ denote the secondary term and $M$ denote the middle term. The four possible '''figures''' are as follows: === $\text I$ === {{:Definition:Figure of Categorical Syllogism/I}} === $\text {II}$ === {{:Definition:Figure of Categorical Syllogism/II}} === $\text {III}$ === {{:Definition:Figure of Categorical Syllogism/III}} === $\text {IV}$ === {{:Definition:Figure of Categorical Syllogism/IV}}


Definition:Figure of Categorical Syllogism/I

{| |- |   Major Premise:   | align="right" | $\map {\mathbf \Phi_1} {M, P}$ |- |   Minor Premise:   | align="right" | $\map {\mathbf \Phi_2} {S, M}$ |- |   Conclusion:   | style="border-style: solid; border-width: 1px 0 0 0"  align="right" | $\map {\mathbf \Phi_3} {S, P}$ |}


Definition:Figure of Categorical Syllogism/II

{| |- |   Major Premise:   | align="right" | $\map {\mathbf \Phi_1} {P, M}$ |- |   Minor Premise:   | align="right" | $\map {\mathbf \Phi_2 } {S, M}$ |- |   Conclusion:   | style="border-style: solid; border-width: 1px 0 0 0"  align="right" | $\map {\mathbf \Phi_3} {S, P}$ |}


Definition:Figure of Categorical Syllogism/III

{| |- |   Major Premise:   | align="right" | $\map {\mathbf \Phi_1} {M, P}$ |- |   Minor Premise:   | align="right" | $\map {\mathbf \Phi_2} {M, S}$ |- |   Conclusion:   | style="border-style: solid; border-width: 1px 0 0 0"  align="right" | $\map {\mathbf \Phi_3} {S, P}$ |}


Definition:Figure of Categorical Syllogism/IV

{| |- |   Major Premise:   | align="right" | $\map {\mathbf \Phi_1} {P, M}$ |- |   Minor Premise:   | align="right" | $\map {\mathbf \Phi_2} {M, S}$ |- |   Conclusion:   | style="border-style: solid; border-width: 1px 0 0 0"  align="right" | $\map {\mathbf \Phi_3} {S, P}$ |}


Definition:Standard Instance of Categorical Syllogism

For each categorical syllogism, there are four figures: :$\begin{array}{r|rl} \text I & & \\ \hline \\ \text{Major Premise}: & \mathbf \Phi_1 & \tuple {M, P} \\ \text{Minor Premise}: & \mathbf \Phi_2 & \tuple {S, M} \\ \hline \\ \text{Conclusion}: & \mathbf \Phi_3 & \tuple {S, P} \\ \end{array} \qquad \begin{array}{r|rl} \text {II} & & \\ \hline \\ \text{Major Premise}: & \mathbf \Phi_1 & \tuple {P, M} \\ \text{Minor Premise}: & \mathbf \Phi_2 & \tuple {S, M} \\ \hline \\ \text{Conclusion}: & \mathbf \Phi_3 & \tuple {S, P} \\ \end{array}$ :$\begin{array}{r|rl} \text {III} & & \\ \hline \\ \text{Major Premise}: & \mathbf \Phi_1 & \tuple {M, P} \\ \text{Minor Premise}: & \mathbf \Phi_2 & \tuple {M, S} \\ \hline \\ \text{Conclusion}: & \mathbf \Phi_3 & \tuple {S, P} \\ \end{array} \qquad \begin{array}{r|rl} \text {IV} & & \\ \hline \\ \text{Major Premise}: & \mathbf \Phi_1 & \tuple {P, M} \\ \text{Minor Premise}: & \mathbf \Phi_2 & \tuple {M, S} \\ \hline \\ \text{Conclusion}: & \mathbf \Phi_3 & \tuple {S, P} \\ \end{array}$ where $\mathbf \Phi_1$, $\mathbf \Phi_2$ and $\mathbf \Phi_3$ each denote one of the categorical statements $\mathbf A$, $\mathbf E$, $\mathbf I$ or $\mathbf O$. A '''standard instance of a categorical syllogism''' is obtained by substituting $\mathbf A$, $\mathbf E$, $\mathbf I$ or $\mathbf O$ for each of $\mathbf \Phi_1$, $\mathbf \Phi_2$ and $\mathbf \Phi_3$ in one of the above figures. Not all of these '''standard instances''' are valid.


Definition:Length of Curve

The '''length''' of a curve is defined as the limit of the length of a polygonal line inscribed within the curve as the maximum length of the chords which form that polygonal line tends to zero.


Definition:Excluded Point Topology/Finite

Let $S$ be finite. Then $\tau_{\bar p}$ is a '''finite excluded point topology''', and $\struct {S, \tau_{\bar p} }$ is a '''finite excluded point space'''.


Definition:Excluded Point Topology/Countable

Let $S$ be countably infinite. Then $\tau_{\bar p}$ is a '''countable excluded point topology''', and $\struct {S, \tau_{\bar p} }$ is a '''countable excluded point space'''.


Definition:Excluded Point Topology/Uncountable

Let $S$ be uncountable. Then $\tau_{\bar p}$ is an '''uncountable excluded point topology''', and $\struct {S, \tau_{\bar p} }$ is an '''uncountable excluded point space'''.


Definition:Excluded Point Topology/Infinite

Let $S$ be infinite. Then $\tau_{\bar p}$ is an '''infinite excluded point topology''', and $\left({S, \tau_{\bar p}}\right)$ is a '''infinite excluded point space'''.


Definition:Countable Complement Topology/Countable

It is possible to define the countable complement topology on a countable set set $S$, but as every subset of a countable set has a countable complement, it is clear that this is trivially equal to the discrete space. This is why the countable complement topology is usually understood to apply to uncountable sets only.


Definition:Particular Point Topology/Finite

Let $S$ be finite. Then $\tau_p$ is a '''finite particular point topology''', and $\struct {S, \tau_p}$ is a '''finite particular point space'''.


Definition:Particular Point Topology/Infinite

Let $S$ be infinite. Then $\tau_p$ is an '''infinite particular point topology''', and $\left({S, \tau_p}\right)$ is an '''infinite particular point space'''.


Definition:Particular Point Topology/Countable

Let $S$ be countably infinite. Then $\tau_p$ is a '''countable particular point topology''', and $\struct {S, \tau_p}$ is a '''countable particular point space'''.


Definition:Particular Point Topology/Uncountable

Let $S$ be uncountable. Then $\tau_p$ is an '''uncountable particular point topology''', and $\struct {S, \tau_p}$ is an '''uncountable particular point space'''.


Definition:Finite Complement Topology/Finite

It is possible to define the finite complement topology on a finite set $S$, but as every subset of a finite set has a finite complement, it is clear that this is trivially equal to the discrete space. This is why the finite complement topology is usually understood to apply to infinite sets only.


Definition:Finite Complement Topology/Countable

Let $S$ be countably infinite. Then $\tau$ is a '''finite complement topology on a countable space''', and $\struct {S, \tau}$ is a '''countable finite complement space'''.


Definition:Finite Complement Topology/Uncountable

Let $S$ be uncountable. Then $\tau$ is a '''finite complement topology on an uncountable space''', and $\struct {S, \tau}$ is a '''uncountable finite complement space'''.


Definition:Fort Space/Countable

Let $S$ be countably infinite. Then $\tau_p$ is a '''countable Fort topology''', and $\struct {S, \tau_p}$ is a '''countable Fort space'''.


Definition:Fort Space/Uncountable

Let $S$ be uncountable. Then $\tau_p$ is an '''uncountable Fort topology''', and $\struct {S, \tau_p}$ is an '''uncountable Fort space'''.


Definition:Complete Metric Space/Definition 1

A metric space $M = \struct {A, d}$ is '''complete''' {{iff}} every Cauchy sequence is convergent.


Definition:Complete Metric Space/Definition 2

A metric space $M = \struct {A, d}$ is '''complete''' {{iff}} the intersection of every nested sequence of closed balls whose radii tend to zero is non-empty.