Proofs Using Basic Equivalences

We can use the basic equivalences to develop proofs as an alternative to using truth tables.

Note: Always place your performed equivalence on a new line, with the name of the equivalence in sideways carrots (<>)

Example: Starting with $(p\wedge q)⟹p$

$\equiv$ <Implication Simp>

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

$\equiv$ <DeMorgan’s>

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

$\equiv$ <Associative Law>

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

$\equiv$ <Commutative>

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

$\equiv$ <Associative>

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

$\equiv$ <Excluded Middle>

T $\vee \neg q$

$\equiv$ <Domination>

$T$

Tautology