Proving logical equivalences

Question

The question is to prove

$\neg (p \wedge q) \to (p \vee r)$ equivalent to $p \vee r$

So far, I got

• $¬[¬(p \wedge q)] \vee (p \vee r)$ - implication
• $(p \wedge q) \vee (p \vee r)$ - double negation

Now, is this question logically not equivalent?

Or is there some way I can prove this is logically equivalent?

Answer 1

Use association, to get $((p\wedge q)\vee p) \vee r$, then use absorption.

Answer 2

$(p \land q) \lor (p \lor r)$ is logically equivalent to $(p \land q) \lor p \lor r$ (the parentheses can be removed because we have the same $\lor$ sign inside and outside the parentheses. This is in turn logically equivalent to $p \land q \lor r$ which implies $p \lor r$. But the converse is not true, so I don't think the question has specified a valid logical equivalence.

Answer 3

To prove the conditional statement: $\neg$ (p $\wedge$ q) $\to$ (p $\vee$ r) you assume the first part and then you could try using De Morgan's Laws, they are listed below.

$\neg$ (p $\wedge$ q) = ($\neg$ p) $\vee$ ($\neg$q)
$\neg$ (p $\vee$ q) = ($\neg$ p) $\wedge$ ($\neg$q)