**Number Systems:**

- $\mathbb{N}$: The set of all
**natural numbers**, i.e., $\mathbb{N}=\left\{0, 1, 2, 3, \ldots\right\}$. - $\mathbb{Z}$: The set of all
**integers**(whole numbers: positive, negative, and zero), i.e., $\mathbb{Z}=\left\{\ldots , -3, -2, -1, 0, 1, 2, 3, \ldots\right\}$ - $\mathbb{Q}$: The set of all
**rational numbers**(numbers that can be expressed as quotients $m/n$ of integers, where $n \neq 0$), i.e., $\mathbb{Q}=\left\{\displaystyle\frac{a}{b}\in\mathbb{R}:a,b\in\mathbb{Z},\ b\ne 0\right\}$ - $\mathbb{R}$: The set of all
**real numbers**, i.e., the numbers representing all points on the real number line. - $\mathbb{C}$: The set of all
**complex numbers**, i.e., $\mathbb{C}=\left\{a + bi:a\in\mathbb{R},\ b\in\mathbb{R}\right\}$.

**Relation**: Let $A$ and $B$ be sets. A **relation** $\mathcal{R}$ between $A$ and $B$ is a subset of the Cartesian product $A\times B$.

**Function**: A **function** $f$ mapping A to B is a relation from A to B that satisfies the following property,

"$\forall a \in A, a$ appears as the first member of exactly one ordered pair in $f$."

**Injective (1-1)**: A function ($f$: $A\to B$) is **injective** if $\forall x \in A\forall y \in A(f(x) = f(y) \to x = y)$.

**Surjective (onto)**: A function ($f$: $A\to B$) is **surjective** if $\forall y \in B\exists x \in A (f(x) = y)$.

**Bijective**: A function $f$ is said to be **bijective** if it is both injective and surjective.

**Cardinality**: The number of elements in a set $A$ is the **cardinality** of $A$, denoted by $|A|$. Two sets $A$ and $B$ have the same **cardinality** if there exists a bijective function from $A$ to $B$.

**Partition**: A **partition** of a set $S$ is a collection of nonempty subsets of $S$ such that every element of $S$ is in exactly one of the subsets of the partition. The subsets are the **cells** of the partition.

**Equivalence Relation**: A relation $\mathcal{R}$ on a set $\mathbf{A}$ is called an **equivalence relation** if the following hold:

- $\forall x\in\mathbf{A}$ $(x\mathcal{R}x)$ (Reflexive Property)
- $\forall x \in\mathbf{A}\forall y \in \mathbf{A}$ $(x\mathcal{R}y \to y\mathcal{R}x)$ (Symmetric Property)
- $\forall x \in\mathbf{A} \forall y \in\mathbf{A} \forall z\in\mathbf{A}$ $(x\mathcal{R}y$ and $y\mathcal{R}z \to x\mathcal{R}z)$ (Transitive Property)

**Binary Operation**: Let $\mathbf{S}$ be a set. A **binary operation** on $\mathbf{S}$ is a function $f$ taking pairs of $\mathbf{S}$ to one element of $\mathbf{S}$. ($f$: $\mathbf{S}$ x $\mathbf{S} \to \mathbf{S}$)

**Commutative**: A binary operation * on a set $\mathbf{S}$ is **commutative** if (and only if) a*b=b*a for all $a, b\in\mathbf{S}$.

**Associative**: A binary operation * on a set $\mathbf{S}$ is **associative** $\iff \forall \mathbf{a} \in \mathbf{S} \forall \mathbf{b} \in \mathbf{S} \forall \mathbf{c} \in \mathbf{S},$ $\mathbf{(a*b)*c = a*(b*c)}$.

**Isomorphism**: An **isomorphism** is a one-to-one (injective) and onto (surjective) function $\gamma$: F $\rightarrow$ G for the binary structures (F, $\ast$) and (G, $\star$), and it satisfies the homomorphism property $\gamma$(a$\ast$b) = $\gamma$(a) $\star$ $\gamma$(b) for all a,b $\in$ F.

**Structural Property**: We say a property of $(\mathbf{S},*)$ is a **structural property** if and only if the property is unchanged when passed through isomorphism.

**Identity Element**: Let $(\mathbf{S},*)$ be a binary structure. An element $e$ of $\mathbf{S}$ is an **identity element** for $*$ if $e*s=s*e=s$ for all $s \in \mathbf{S}$.

i.e., $\exists e \in \mathbf{S} \forall s \in \mathbf{S}(e*s=s*e=s).$

**Inverse**: Let $(\mathbf{S},*)$ be a binary structure with identity element $e$. If for all $s \in \mathbf{S}$ there is an element $s^{-1} \in \mathbf{S}$ such that $s*s^{-1}=s^{-1}*s=e$, then $s^{-1}$ is the **inverse** of $s.$ i.e., $\forall s \in \mathbf{S} \exists s^{-1} \in \mathbf{S} (s*s^{-1}=s^{-1}*s=e).$

**Group**: A **group** $\langle \textit{G}, * \rangle$ is a set $\textit{G}$, closed under a binary operation *, such that the following axioms are satisfied:

$\mathcal{G}_1$: For all $\textit{a, b, c,} \in \textit{G}$, we have

$(\textit{a}*\textit{b})*\textit{c}=\textit{a}*(\textit{b}*\textit{c}).$$\textbf{associativity of *}$

$\mathcal{G}_2$: There is an element $\textit{e}$ in $\textit{G}$ such that for all x $\in \textit{G}$.

$\textit{e} *\textit{x} = \textit {x} * \textit {e}=\textit {x}$. $\textbf{identity element} \textit {e} for *.$

$\mathcal{G}_3$: Corresponding to each $\textit{a}\in\textit{G}$, there is an element $\textit {a} ^\prime$ in$\textit{G}$ such that

$\textit{a}*\textit{a} ^\prime=\textit{a} ^\prime*\textit{a}=\textit{e}. \textbf{inverse} \textit{a} ^\prime of \textit{a}$.

**Abelian**: A group G is **abelian** if its binary operation is commutative.

**$GL(n,\mathbb{R})$**: The general linear group of degree n is the set of n×n invertible matrices with real coefficient values. This forms a group under matrix multiplication.

**Subgroup**: Let $\mathbf{G}$ be a group. We say a subset $\mathbf{H} \subseteq \mathbf{G}$ is a **subgroup** if $\mathbf{H}$ is closed under the binary operation from $\mathbf{G}$ and $\mathbf{H}$ is also a group.

**Proper Subgroup**: a proper subgroup of a group $G$ is a subgroup $H$ of $G$ $|$ $H\neq G$ written $H\subset G$.

**Order of a group**: The order $|G|$ of a group $G$ is the number of elements in $G$.

**Order of an element**: Let $G$ be a group. The order of an element $c \in G$ is $|<c>|$. If it is of finite order $m$, then $m$ is the smallest integer such that $c^m=e$. Otherwise it is of infinite order.

**Cyclic group**:Let $\textit{G}$ be a group. We say $\textit{G}$ is $\mathbf{cyclic}$ if there exists an element $\textit{a} \in \textit{G}$ with $\langle \textit{a}\rangle$ = $\textit{G}$ where $\langle a\rangle$ = {$a^n | \textit{n} \in \mathbb{Z}$}. Since $\langle \textit{a}\rangle$ = $\textit{G}$, we say $\textit{a}$ generates G.

**Generator**: Let $\textit{G}$ be a group. If $\langle \textit{a}\rangle$ = $\textit{G}$, we say $\textit{a}$ generates $\textit{G}$, or $\textit{a}$ is a **generator** for $\textit{G}$. Note: $\langle \textit{a}\rangle$ := {$a^n | \textit{n} \in \mathbb{Z}$} is called the cyclic subgroup generated by $\textit{a}$.

**Division Algorithm**:Let $n\in\mathbb{Z}$ and $m\in\mathbb{Z}^+$. Then $n=mq+r$ where $q\in\mathbb{Z}$ is unique, and $0\leq r<m$ is unique.

**Greatest Common Divisor**:Let $r$ and $s$ be two positive integers. The positive generator $d$ of the cyclic group

$H=\left\{nr+ms|n,m\in \mathbb{Z}\right\}$

under addition is the **greatest common divisor** (abbreviated gcd) of $r$ and $s$. We write $d=$gcd$(r,s)$.

**Relatively Prime**: Two positive integers are **relatively prime** if their greatest common divisor is 1.

**Permutation**: A **Permutation of a set** *A* is a function $\phi : A \rightarrow A$ that is both one to one and onto.

**Orbit of Permutation**: Define an equivalence relation $\sim$ such that for $a,b \in A$, let $a \sim b$ if and only if $b=\sigma^n(a)$ for some $n \in \mathbb{Z}.$ Let $\sigma$ be a permutation of a set $A$. The equivalence classes in $A$ determined by the equivalence relation $\sim$ are the **orbits of $\sigma$**.

**Cycle Permutation**: A permutation $\sigma \in S_n$ is a **cycle** if it has at most one orbit containing more than one element. The **length** of a cycle is the number of elements in its largest orbit.

**Transposition**: A cycle of length 2 is called a **transposition**.

**Even Permutation**: A permutation is an **Even Permutation** iff it can be expressed as the composition of an even number of transpositions.

**Left Coset**: Let $H$ be a subgroup of a group $G$. The subset $aH = \left\{ah|h \in H\right\}$ of $G$ is the **left coset** of $H$ containing $a$.

**Index**: Let H be a subgroup of a group G. The number of left cosets of H in G is the **index** (G:H) of H in G.

**Direct Product of Groups**: If $(G, \ast)$ and $(K, \star)$ are both groups, we can create a new group called **the direct product of $G$ and $K$** by using $(G \times K, (\ast, \star))$. So $G \times K$ is a group with "component-wise operation."

**Homomorphism**: Let $(G, \ast)$ and $(G\prime, \star)$ be groups. We say a function $\varphi: G \rightarrow G\prime$ is a **homomorphism** if and only if $\varphi(a \ast b) = \varphi(a) \star \varphi(b)$ for all a $\in G$ and b \in G$.

**Kernel**: Let $\phi : G \rightarrow G'$ be a homomorphism of groups. The subgroup $\phi^{-1}[\left\{e'\right\}] = \left\{ x \in G | \phi(x) = e'\right\}$is the **kernel of** $\phi$, denoted by $Ker(\phi)$

**Image**: Let $\varphi : X \rightarrow Y$ be a function and $A \subseteq X$ and $B \subseteq Y$. The image of $A$ is $\varphi[A] = \{y \in Y | \varphi(a) = y$ for some $a \in A\}$.

**Normal Subgroup**:A subgroup $H$ of a group $G$is **normal** if its left and right cosets coincide, that is, if $gH=Hg$ for all $g \in G$

**Factor Group (or Quotient Group)**: Let $G$ be a group and $H \triangleleft G$. Then the left cosets of $H$ in $G$ (denoted $G/H$) is a group under coset multiplication and is called the factor or quotient group.