Double Conditional Equivalence Relation

Question

For all $a \in\mathbb{Z}$, $a \sim a + 3$;

For all $a \in\mathbb{Z}$, $a \sim a + 7$;

Show that $a ∼ b$ for all $a,b \in\mathbb{Z}$.

I found the equivalence above online yet I don't comprehend how this relation is equivalent for $a\sim b$.

Thank you for any guidance or help given.

Answer 1

Hint 1: For every $a\in\mathbb Z$,

$$a+1 = a+7-3-3\\ a-1=a+3+3-7$$

Hint 2:

For every $a$, you can also show that $a\sim a-7$ and $a\sim a-3$