Show that $0 \in H$. Show that if $a \in H$ then $-a \in H$, and lastly, if $a$ and $b \in H$ then $a + b ∈ H$.
Last bonus question on our last unit test on sets, equivalence relations and proofs. I wasn't able to even start this question like the last few and was hoping for some help. Please consider that this is a proofs course and proper proof procedure is required in all answers.
I'm thinking the question states $a \sim b \iff (a - b) \in H$, in which case,
We know the set $H$ is non-empty and hence it has an element $a$. Since it is an equivalence relation $a$ is related to $a$. Hence $a \sim a \implies (a - a) = 0 \in H$. Suppose $a \in H.$ Then $ (a - 0) \in H$, then $a \sim 0$. By reflexivity $0 \sim a$ and hence $(0 - a) = -a \in H$.
Finally, $(a + 2b) - 2b = a \in H \implies (a + 2b) \sim 2b$ and $2b - b = b \in H \implies 2b \sim b$. By transitivity, $(a + 2b )\sim b \implies (a + 2b) - b = (a + b) \in H$