Answer
See step by step work for answer.
Work Step by Step
Two propositions are logically equivalent if they have the same truth value for any combination of truth values of the variables.
A conjunction p$\lor $q is true, if either of p or q are true.
A negation ¬p is true if p is false.
p $\downarrow$ q is true if and only if both p and q are false.
$\underline{p \quad q \quad p \lor q \quad \neg(p\lor q) \quad p\downarrow q}$
$T \quad T \quad T \quad \quad \quad F \quad \quad \quad F$
$T \quad F \quad T \quad \quad \quad F \quad \quad \quad F$
$F \quad T \quad T \quad \quad \quad F \quad \quad \quad F$
$F \quad F \quad F \quad \quad \quad T \quad \quad \quad T$
Since the last two columns of the truth table contain the same truth value in every row, the last two expressions are logically equivalent.
Hence, $p\downarrow q\equiv \neg(p\lor q)$
