With the aid of Truth tables, determine which of the following logical statements (or expressions) is a Tautology, a Contradiction, or a Contingency. (a) (p ↔ q) ↔ ((p → q) ∧ (q → p)) (b) (p → (q → r)) ↔ (p ∧ q → r) (c) (p → r ∨ q)⊕(q ∧ r) (d) p ∨ q ∧ (¬r → ¬p ∧ ¬q) 2. With reference to the following logical equivalences, determine whether each of them is valid or invalid. If it is invalid then give a counterexample (e.g. based on a Truth-Value assignment). If it is valid then give an algebraic proof using ONLY logical equivalences from Table 6 in Section 1.3 of course textbook, and the rule for conditional: p → q ≡ ¬p ∨ q. (a) p → (q → r) ≡ (p → q) → r (b) (p → r) ∨ (q → r) ≡ (p ∧ q) → r (c) (¬p → (q ∧ ¬q)) ≡ p (d) p ∧ q → r ≡ (¬p ∧ ¬q) → ¬r 3. Express the following logical statements in terms of the Contrapositive, Converse, and Inverse, respectively. (a) You are an adult provided that you are above 18 years of age. (b) It is daylight whenever the sun is shining. (c) 3 × 2 = 6 only if 2 × 3 = 6.
1. With the aid of Truth tables, determine which of the following logical statements (or expressions)
is a Tautology, a Contradiction, or a Contingency.
(a) (p ↔ q) ↔ ((p → q) ∧ (q → p))
(b) (p → (q → r)) ↔ (p ∧ q → r)
(c) (p → r ∨ q)⊕(q ∧ r)
(d) p ∨ q ∧ (¬r → ¬p ∧ ¬q)
2. With reference to the following logical equivalences, determine whether each of them is valid or
invalid. If it is invalid then give a counterexample (e.g. based on a Truth-Value assignment).
If it is valid then give an algebraic proof using ONLY logical equivalences from Table 6 in
Section 1.3 of course textbook, and the rule for conditional: p → q ≡ ¬p ∨ q.
(a) p → (q → r) ≡ (p → q) → r
(b) (p → r) ∨ (q → r) ≡ (p ∧ q) → r
(c) (¬p → (q ∧ ¬q)) ≡ p
(d) p ∧ q → r ≡ (¬p ∧ ¬q) → ¬r
3. Express the following logical statements in terms of the Contrapositive, Converse, and Inverse,
respectively.
(a) You are an adult provided that you are above 18 years of age.
(b) It is daylight whenever the sun is shining.
(c) 3 × 2 = 6 only if 2 × 3 = 6.
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