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Group $(G, *)$DefinitionLet $*$ be a binary operation on $G$.$$*:G\times G \to G$$ Binary operation $*: G\times G \to G$associative$a*(b*c)=(a*b)*c$ for $\forall\ a, b, c\in G$identity$\exists e\in G$ such that $\ a*e=e*a=a$ for $\forall\ a\in G$inverse$\exists a^{-1}\in G$ such that $\ a* a^{-1} = a^{-1}*a =e$ for $\forall\ a\in G$If $G$ sufficies the above conditons, it is called a group under..

Equivalence relationDefinitionLet $A$ be a non-empty set.A binary relation $~$ on $A$ is an equivalence relation if for every $x, y, z$ in $A$, it suffices the following properties:(reflexive) $x~x$.(symmetric) if $x~y$, then $y~x$.(transitive) if $x~y$ and $y~z$, then $x~z$. ExampleShow that congruence modulo $n$ is an equivalent relation. Note that $a$ be congruent to $b$ modulo $n$ where $a, ..

Congruence RelationDefinitionFor $a, b \in \mathbb{Z}$ and $n \in \mathbb{Z}_+$, we say that $a$ is congruent to $b$ modulo $n$ if there exists $k \in \mathbb{Z}$ such that$$nk=b−a$,$nk = b - a$$,which is denoted by$$a \equiv b \pmod{n}$$. DefinitionFor $n \in \mathbb{Z}_+$, the set$$\{0, 1, \dots, n - 1\}$$is called the complete system of residues modulo $n$. TheoremLet $a, b, c \in \mathbb{Z}$..

Finite and inifite setsDefinitionA set $A$ is finite if $A = \emptyset$ or there is a bijection $f: A \to {1, 2, \cdots, n}.$If $A$ is not finite, then $A$ is infinite. Countability of setsDefinitionIf there is a bijection $f: A \to B$, $A$ is cardinally equivalent to $B$. DefinitionIf a set $A$ is cardinally equivalent to $\mathbb{N}$, then $A$ is countably inifinite.If a set $A$ is either fini..