Propositional Equivalences Practice Questions

Propositional equivalences are logical relationships in which two propositions always have the same truth value for every possible combination of truth values.

Question 1: Show that p ∧ (p ∨ q) ≡ p
Solution:

p ∧ (p ∨ q)
≡ (p ∧ p) ∨ (p ∧ q) (Distributive Law)
≡ p ∨ (p ∧ q) (Idempotent Law)
≡ p (Absorption Law)

Question 2: Prove that (p → q) ∧ (p → r) ≡ p → (q ∧ r)
Solution:

(p → q) ∧ (p → r)
≡ (¬p ∨ q) ∧ (¬p ∨ r) (Definition of Implication)
≡ ¬p ∨ (q ∧ r) (Distributive Law)
≡ p → (q ∧ r) (Definition of Implication)

Question 3: Show that ¬(p ↔ q) ≡ p ↔ ¬q
Solution:

¬(p ↔ q)
≡ ¬((p → q) ∧ (q → p)) (Biconditional)
≡ ¬(p → q) ∨ ¬(q → p) (De Morgan)
≡ ¬(¬p ∨ q) ∨ ¬(¬q ∨ p) (Implication)
≡ (p ∧ ¬q) ∨ (¬p ∧ q) (De Morgan)
≡ p ⊕ q (XOR)
≡ p ↔ ¬q (Equivalence)

Question 4: Prove that p ∨ (¬p ∧ q) ≡ p ∨ q
Solution:

p ∨ (¬p ∧ q)
≡ (p ∨ ¬p) ∧ (p ∨ q) (Distributive Law)
≡ T ∧ (p ∨ q) (Law of Excluded Middle)
≡ p ∨ q (Identity Law)

Question 5: Show that (p ∧ q) → r ≡ p → (q → r)
Solution:

(p ∧ q) → r
≡ ¬(p ∧ q) ∨ r (Definition of Implication)
≡ (¬p ∨ ¬q) ∨ r (De Morgan's Law)
≡ ¬p ∨ (¬q ∨ r) (Associative Law)
≡ ¬p ∨ (q → r) (Definition of Implication)
≡ p → (q → r) (Definition of Implication)

Practice Questions

Question 1. Prove that p → (q → r) ≡ (p ∧ q) → r.

Question 2. Show that (p → q) ∨ (p → r) ≡ p → (q ∨ r).

Question 3. Demonstrate that ¬(p → q) ≡ p ∧ ¬q.

Question 4. Prove that (p → q) ∧ (p → ¬q) ≡ ¬p.

Question 5. Show that (p ∨ q) ∧ (p ∨ r) ≡ p ∨ (q ∧ r).