A group $(G,*)$ is abelian if $*$ is commutative.
A line, line segment, or ray is an angle bisector of an angle if it divides the angle into two equal angles.
A binary operation $*$ on $S$ is associative if $\forall a,b,c \in S, (a*b)*c = a*(b*c)$.
A function $f: A \to B$ is a bijection if $f$ is injective and surjective. If $f$ is a bijection, then we say $f$ is bijective.
A function $*:A \times B \to C$ is a binary operation. Usually it is denoted like $a*b$ instead of $*(a,b)$ like other functions. If $A=B=C$ then $*$ is a binary operation on $A$.
If you circumscribe a circle around a triangle, then the circle is called the circumcircle of the triangle. The center of the circumcircle is called the circumcenter. The radius of the circumcircle is called the circumradius.
Let $(X,\mathcal{T})$ be a topological space. Then a subset $F$ of $X$ is closed in $\mathcal{T}$ if $X \setminus F$ is open in $\mathcal{T}$.
A binary operation $*$ defined on $S \times S$ is commutative if $\forall a,b \in S, a*b = b*a$.
A ring $(R,+,*)$ is a commutative ring if $*$ is commutative.
A group $(G,*)$ is cyclic if $\exists g \in G$ such that $G = \{g^n : n \in \mathbb{Z} \}$, where $g^n$ could be the identity element if $n = 0$, $g$ multiplied by itself $n$ times if $n$ is positve, or $g^{-1}$ multiplied by itself $-n$ times if $n$ is negative. We say that $G$ is generated by $g$ and $\langle g \rangle = G$.
If $(X_1, \mathcal{T}_1)$ and $(X_2, \mathcal{T}_2)$ are topological spaces, then a map $f: X_1 \to X_2$ is continuous if for every $U \in \mathcal{T}_2, f^{-1}(U) \in \mathcal{T}_1$.
A permutation $\sigma : S \to S$ is a derangement if $\forall x \in S, \sigma (x) \neq x$. In other words, there are no fixed points.
A set and two binary operations $(F,+,*)$ constitute a field if they have these properties:
Closure:
1) $\forall a,b \in F, a + b \in F$ and $a * b \in F$.
Associativity:
2) $+$ and $*$ are associative.
Commutativity:
3) $+$ and $*$ are commutative.
Identity Elements:
4) $\exists 0 \in F$ such that $\forall a \in F, a + 0 = a$. $0$ is called the additive identity, but it is not necessarily the number 0.
5) $\exists 1 \in F$ such that $\forall a \in F, a * 1 = a$. $1$ is called the multiplicative identity, but it is not necessarily the number 1.
6) The additive identity $0$ and multiplicative identity $1$ are distinct.
Inverse Elements:
7) $\forall a \in F, \exists a' \in F$ such that $a + a' = 0$. $a'$ is called the additive inverse of $a$ and is usually denoted as $-a$.
8) $\forall a \in F$ except $0, \exists a' \in F$ such that $a * a' = 1$. $a'$ is called the multiplicative inverse of $a$ and is usually denoted as $a^{-1}$.
Distributivity:
9) $\forall a,b,c \in F, a*(b+c) = a*b + a*c$.
Let $(G,+, \star)$ and $(H, \oplus, \times)$ be fields. Then the function $f: G \to H$ is a homomorphism if $\forall a,b \in G, f(a) \times f(b) = f(a \star b)$ and $f(a) \oplus f(b) = f(a + b)$. This is analagous to a ring homomorphism.
Let $(G,+, \star)$ and $(H, \oplus, \times)$ be fields. Then the function $f: G \to H$ is an isomorphism if $f$ is a bijection and homomorphism. If there exists an isomorphism between the two fields, then they are isomorphic.
A pair of sets $(V,E) = G$ is a graph if
1) $V$ is nonempty.
2) $\forall e \in E, e \subseteq V$ and $|e| = 2$.
We call $V$ the vertex set of $G$. Elements of $V$ are called vertices. We call $E$ the edge set of $G$. Elements of $E$ are called edges.
A set and a binary operation $(G,*)$ constitute a group if they have these properties:
Closure:
1) $\forall a,b \in G, a*b \in G$
Associativity:
2) $*$ is associative.
Identity Element:
3) $\exists e \in G$ such that $\forall a \in G, e*a = a*e = a$. $e$ is called the identity element of $G$.
Inverse Elements:
4) $\forall a \in G, \exists a'$ such that $a*a' = a'*a = e$. $a'$ is called the inverse element of $a$ and is usually denoted as $a^{-1}$.
Let $(G,\star)$ and $(H,\times)$ be groups. Then the function $f: G \to H$ is a homomorphism if $\forall a,b \in G, f(a) \times f(b) = f(a \star b)$.
Let $(G,\star)$ and $(H,\times)$ be groups. Then the function $f: G \to H$ is an isomorphism if $f$ is a bijection and homomorphism. If there exists an isomorphism between the two groups, then they are isomorphic.
A square matrix is an identity matrix if every entry on its diagonal is 1 and every other entry in the matrix is 0. Notice that multiplying the identity matrix with another matrix of the same size does not change the other matrix. The identity matrix is usually represented by the letter $I$.
If you inscribe a circle inside of a triangle, then the circle is called the incircle of the triangle. The center of the incircle is called the incenter. The radius of the incircle is called the inradius.
A function $f: A \to B$ is an injection if $\forall x,y \in A$, $f(x) = f(y) \implies x = y$. If $f$ is an injection, then we say that $f$ is injective.
An integer is a number in the set $\mathbb{Z} = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}$.
A triangle is an isoceles triangle if at least two of the sides of the triangle are equal in length.
A quadrilateral is a kite if two adjacent sides of the quadrilateral are equal in length and the remaining two sides are also equal in length.
A line segment is a median of a triangle if one of its endpoints is on a vertex of the triangle and the other endpoint is on the midpoint of the side opposite to the aforementioned vertex.
A polyomino is a geometric figure in a 2-dimensional plane made by starting with a square and then placing a non-negative number of squares one at a time somewhere such that each square shares an edge with another square without sharing all of the edges with the same square. Two one-sided polyominos are distinct if they can not be rotated or translated onto each other. Also, assume that all the squares in all of the polyominos are the same size.
A quadrilateral is a parallelogram if every side of the quadrilateral is parallel to the opposite side.
A bijection from a set $S$ to itself is a permutation of $S$. Usually, the domain of a permutation has a finite number of elements.
A line is a perpendicular bisector of a line segment if the line is perpendicular to the line segment and the line passes through the midpoint of the line segment.
A number is prime if it is an integer greater than 1 that has no divisors other than 1 and itself.
A quadrilateral is a rhombus if all of the sides are equal in length.
A set and two binary operations $(R,+,*)$ constitute a ring if they have these properties:
Closure:
1) $\forall a,b \in R, a + b \in R$ and $a * b \in R$.
Associativity:
2) $+$ and $*$ are associative.
Commutativity:
3) $+$ is commutative.
Identity Elements:
3) $\exists 0 \in R$ such that $\forall a \in R, a + 0 = a$. $0$ is called the additive identity, but it is not necessarily the number 0.
4) $\exists 1 \in R$ such that $\forall a \in R, a * 1 = 1 * a = a$. $1$ is called the multiplicative identity, but it is not necessarily the number 1.
Inverse Elements:
5) $\forall a \in R, \exists a' \in R$ such that $a + a' = 0$. $a'$ is called the additive inverse of $a$ and is usually denoted as $-a$.
Distributivity:
6) $\forall a,b,c \in R, a*(b+c) = a*b + a*c$ and $(b+c)*a = b*a + c*a$.
Let $(G,+, \star)$ and $(H, \oplus, \times)$ be rings. Then the function $f: G \to H$ is a homomorphism if $\forall a,b \in G, f(a) \times f(b) = f(a \star b)$ and $f(a) \oplus f(b) = f(a + b)$.
Let $(G,+, \star)$ and $(H, \oplus, \times)$ be rings. Then the function $f: G \to H$ is an isomorphism if $f$ is a bijection and homomorphism. If there exists an isomorphism between the two rings, then they are isomorphic.
A set $A$ is a subset of a set $B$ if every element in $A$ is also an element of $B$. This is denoted as $A \subseteq B$.
Two angles $a$ and $b$ are supplementary if they add up to 180 degrees.
A function $f: A \to B$ is a surjection if $\forall y \in B, \exists x \in A$ such that $f(x) = y$. If $f$ is a surjection, then we say that $f$ is surjective.
A set of subsets $\mathcal{T}$ of a set $X$ is a topology on $X$ if
1) The empty set and $X$ are in $\mathcal{T}$.
2) For every $T_0 \subseteq \mathcal{T}$, $\left( \bigcup\limits_{U \in T_0} U \right) \in \mathcal{T}$. That is, any finite or infinite union of elements in $\mathcal{T}$ produces an element in $\mathcal{T}$.
3) For every $U,V \in \mathcal{T}, U \cap V \in \mathcal{T}$. Equivalently, any finite intersection of elements in $\mathcal{T}$ produces an element in $\mathcal{T}$.
If $\mathcal{T}$ is a topology on $X$, then $(X, \mathcal{T})$ is a topological space.