Biconditional statement

Let $p$ and $q$ be propositions. The biconditional statement $p⟺q$, is the proposition that "$p$ if and only if $q$". This is also called an bi-implication or an equivalence.

$$v(A⟺B)=\{\begin{array}{ll}1,& \text{if }v(A)=v(B)\\ 0,& \text{if }v(A)\ne v(B)\end{array}$$

$p$	$q$	$p⟺q$
T	T	T
T	F	F
F	T	F
F	F	T

There are some implicit uses of biconditional statements in English, for example:
- If you finish your meal then you can have desert, $p \rightarrow q$.
 
But we actually mean $p \iff q$:
- You can have desert if and only if you finish your meal.