hcju/proofwikips huggingface dataset:
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
{{: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
{{: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
{{:Definition:Secant Function/Definition from Circle/First Quadrant}}
Definition:Cosecant/Definition from Circle
{{: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''' : :$S \notin S$ That is, $S$ is not an element of itself.
Definition:Abnormal Set
Let $S$ be a set. Then $S$ is an '''abnormal set''' : :$S \in S$ 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)}}
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 $0$.
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'''  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''' : :$f$ has both a left inverse and a right inverse.
Definition:Bijection/Definition 3
A mapping $f: S \to T$ is a '''bijection''' : :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''' : :$(1): \quad$ 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''' : :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 $y$'''  they are both divisors of each other. That is, $x$ and $y$ are '''associates'''  $x \divides y$ and $y \divides x$.
Definition:Composant/Continuum
$C$ is a '''composant''' of $H$ : :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 $p$ : :$C$ 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 $T$'''  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 $T$'''  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.  :''{{: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.