If the Boolean expression (p ⇒ q) ⇔ (q * ( ∼ q) is a tautology, then the boolean expression p * ( ∼ q) is equivalent to :

Question

If the Boolean expression \((p \Rightarrow q)\Leftrightarrow (q\, *(\sim p))\) is a tautology, then the Boolean expression \(p\, *(\sim q)\) is equivalent to :

(1) \(q \Rightarrow p\)

(2) \(\sim q \Rightarrow p\)

(3) \(p \Rightarrow \sim q\)

(4) \(p \Rightarrow q\)

Answer 1

Answer: (a)

(p → q) ⇔ (q * ~ P)

p	q	p → q	q * ~ P	~ q	~ q ∧ p	~ (~ q ∧ p)
T	T	T	T	F	F	T
T	F	F	F	T	T	F
F	T	T	T	F	F	T
F	F	T	T	T	F	T

∴ ~ (~ q ∧ p) = q ∨ ~ p = ~ p ∨ q

∴ * is equivalent to ∨

∴ p * ~ q = p ∨~ q

= ~ q ∨ p

= q⇒ p