Which one of the following Boolean expressions is a tautology ?

Question

Which one of the following Boolean expressions is a tautology ? (A) (P ∨ q) ∧ (∼ P ∨ ∼ q) (B) (P∧ q) ∨ (P ∧ ∼ q) (C) (P ∨ q) ∧ (P∨∼ q) (D) (P ∨ q) ∨ (P∨ ∼ q)

Tardigrade Admin · Accepted Answer

(1) (p ∨ q) ∧ (∼ p ∨∼ q) ≡ (p ∨ q) ∧∼ (p ∧ q) → Not tautology (Take both p and q as T) (2) (p ∧ q) ∨ (p ∧∼ q) ≡ p∧(q ∨∼ q) ≡ p ∧ t ≡ p (3) (p ∨ q) ∧ (p ∨ ∼ q) ≡ p ∨ (q ∧ ∼ q) ≡ p ∨ c ≡ p (4) (p∨ q) ∨ (p ∨ ∼ q) ≡ p ∨(q ∨ ∼ q) ≡ p ∨ t ≡ t

Q. Which one of the following Boolean expressions is a tautology ?

$(P \lor q) \land (\sim P \lor \sim q)$

$(P \land q) \lor (P \land \sim q)$

$(P \lor q) \land (P \lor \sim q)$

$(P \lor q) \lor (P \lor \sim q)$

Q. Which one of the following Boolean expressions is a tautology ?

(P∨q)∧(∼P∨∼q)

(P∧q)∨(P∧∼q)

(P∨q)∧(P∨∼q)

(P∨q)∨(P∨∼q)

(1) (p∨q)∧(∼p∨∼q)≡(p∨q)∧∼(p∧q)→ Not tautology (Take both p and q as T) (2) (p∧q)∨(p∧∼q)≡p∧(q∨∼q)≡p∧t≡p (3) (p∨q)∧(p∨∼q)≡p∨(q∧∼q)≡p∨c≡p (4) (p∨q)∨(p∨∼q)≡p∨(q∨∼q)≡p∨t≡t

$(P \lor q) \land (\sim P \lor \sim q)$

$(P \land q) \lor (P \land \sim q)$

$(P \lor q) \land (P \lor \sim q)$

$(P \lor q) \lor (P \lor \sim q)$