Let R be the relation on integers defined by aRb if and only if 2a+3b congruent to 0 (mod 5). Show that this is an equivalence relation.
ok, symmetry is a bit difficult, i'l admit when seeing this for the first time. so i will do it for you. but i will leave transitivity to you. just express the congruence as i did.
Note that $\displaystyle 2a + 3b \equiv 0~\mbox{mod }5$ means $\displaystyle 2a + 3b = 5k$ for some $\displaystyle k \in \mathbb{Z}$
Claim: R is symmetric
Proof: Assume $\displaystyle aRb$, that is, $\displaystyle 2a + 3b = 5k$ for some integer $\displaystyle k$
Then, $\displaystyle 2b + 3a = 5b - 3b + 5a - 2a$
$\displaystyle \Rightarrow 2b + 3a = 5a + 5b - \underbrace{(2a + 3b)}_{5k}$
$\displaystyle \Rightarrow 2b + 3a = 5(a + b - k)$
since $\displaystyle (a + b - k) \in \mathbb{Z}$, we have $\displaystyle bRa$
Thus, R is symmetric