Please help with this question!

Question

Prove if it is tautology (p→q) ∨ [(p ∨ ¬q ∨ p) ∨ ¬r)] with the use propositional laws.

*please use the propositional laws to prove this.

Answer 1

The given is:

p->q v [ (p v ~q v p) v ~r ] =

associative/commutative:

p->q v [ (~q v (p v p)) v ~r] =

p->q v [ ~q v p v ~r ] =

implication identity:

~p v q v [ ~q v p v ~r ]

associative/commutative:

(~p v p) v (q v ~q) v ~r

X or ~X MUST be true:

True v True v ~r

True