Can someone please verify whether my proofs are logically correct? :)
Proof:
For addition, $a+b\equiv b+a \;\; (mod\;\;n)$ iff there exists an integer $k$ such that $(a +b)-(b+a)=nk$. Since $(a+b)-(b+a)=0$, there exists an integer $k=0$ that satisfies the equation. Therefore, $a+b\equiv b+a \;\; (mod\;\;n)$. $\square$
For multiplication, $ab\equiv ba \;\; (mod\;\;n)$ iff there exists an integer $k$ such that $(ab)-(ba)=nk$. Since $(ab)-(ba)=0$, there exists an integer $k=0$ that satisfies the equation. Therefore, $ab\equiv ba \;\; (mod\;\;n)$. $\square$