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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, ..