Direct Proof of P ⇒ Q

1. Assume LHS is T
2. Argue RHS is T p → q ≡ ¬ q → ¬ p

Indirect Proof (By Contrapositive)

1. Assume RHS is F
2. Argue LHS is F

Therefore, both (p or q) ⊕ (p ∧ q) is T ∧ ¬ p is T. Then, P is F. Therefore, p ^ q is F. Then p or q is T. Means q is T which means q is T since p is F. So the implication is valid.

Then p ∧ r is T. Therefore both p ∧ r are T. So, p or q is T (since p is T). (p or q) ⊕ (p ∧ r) is F. So LHS is F. So the implication is valid.

Assume LHS is T

1. p or q T ∧ (p ∧ r | F)
2. p or q | T | ∧ p ∧ r | F

Both p or r is T ∧ p ⊕ q is T. Want to show q → r is T. So assume q is T. Since p ⊕ q is T, p is F. Since p or r is T, so r is T which is what we needed to show. So RHS is T.

Implication is Valid

Assume RHS is F. So q → r is F. means q is T ∧ r is F.

1. p T: Since q is also T p ⊕ q is F then LHS is F
2. p F: Since r is also F p or is also F LHS is F