Use the laws of propositional logic to prove p ↔ (p ∧ r) ≡ ¬p ∨ r.

Question

Use the laws of propositional logic to prove p ↔ (p ∧ r) ≡ ¬p ∨ r

Answer 1

First of all, the proposition to be proved isn't well-formed; it needs more parentheses to clearly indicate the scope of the operators involved. In other words, the proposition in question needs to be specified as one of the following:

[p ↔ (p ∧ r)] ≡ (¬p ∨ r)

p ↔ [(p ∧ r) ≡ (¬p ∨ r)]

p ↔ {[(p ∧ r) ≡ ¬p] ∨ r}

{p ↔ [(p ∧ r) ≡ ¬p]} ∨ r

Once the proposition has been sufficiently specified, how we prove it depends on the laws/rules we have. I imagine that the proposition that must be proved is one of the first two options, which are both biconditionals. If so, then it can be proved by using indirect proof or two conditional proofs. To use indirect proof, assume the negation of the proposition to be proved and then use whatever rules you have to derive a contradiction (how this contradiction is derived of course depends on the rules at your disposal). To use two conditional proofs, first prove that the left side of the biconditional implies the right side, and then prove that the right side implies the left side. That is: first assume the left side and derive the right side, then assume the right side and derive the left side. These two conditional proofs then entail the biconditional to be proved. And again, how these proofs proceed will depend on the rules at your disposal, so we can't do the proof until we have these rules.