How to prove a reduction
Follow the definition of $\leq_p$.

You have to find a function $f$ (define it!), computable in polynomial time (prove this!) such that

$$
\begin{array}{l}
\forall x.\  x\in A \implies f(x)\in B
\\
\forall x.\ x\notin A \implies f(x)\notin B
\end{array}
$$
(two more proofs!)

More intuitively, $f$ converts instances of $A$ (i.e. $x$) into instances for $B$ ($f(x)$), and must map $A$ inside $B$, and the complement of $A$ inside the complement of $B$. Very, very roughly, it describes how to solve $A$ exploiting $B$.

Finally, never write nonsense formulas like $A \implies B$ since sets do not imply other sets. Try to be as formal as you can, at least when you have not yet mastered the definition.