Equivalence-Style Proofs

Use Basic Equivalences to manipulate compound propositions.

We can also use them to prove things:

• compound propositions are tautologies
• one compound proposition is equivalent to another

Very similar layout and process to proofs using basic equivalences.

Example: $p⟹q\equiv \neg q⟹\neg p$ Contrapositive (basic equivalence)

Start on one side and manipulate it until you hopefully end up with the other side

Preferably start with the more complicated one.

$\neg q⟹\neg p$ $\equiv$ <Implication Simplification> $\neg (\neg q)\vee \neg p$ $\equiv$ <Double Negation> $q\vee \neg p$

Trying to get to $p⟹q$

$\equiv$ <Commutativity> $\neg p\vee q$ $\equiv$ <Implication Simplification> $p⟹q$

Therefore, we have $p⟹q\equiv \neg q⟹\neg p$