“A implies B”, written symbolically as $A⟹B$, is defined by the truth table:

\hline A & B & A \implies B\\ \hline T & T & T\\ T & F & F\\ F & T & T\\ F & F & T\\ \hline \end{array}

$A$ is referred to as the hypothesis for the implication, and $B$ is referred as the conclusion.

The Inverse

The implication $(\neg A)⟹(\neg B)$ is the inverse of $A⟹B$

The Converse

The implication $B⟹A$ is the converse of $A⟹B$

The Contrapositive

The implication $(\neg B)⟹(\neg A)$ is the contrapositive of $A⟹B$ $(A⟹B)\equiv ((\neg B)⟹(\neg A))\text{(logically equivalent)}$

Other remarks

$((A\vee B)⟹C)\equiv ((A⟹C)\wedge (B⟹C))$ $(A⟹B)\equiv ((\neg A)\vee B)$ $\neg (A⟹B)\equiv (A\wedge (\neg B))\text{(negation of implication)}$