Question

Negation of the Boolean expression `p Leftrightarrow (q \implies p)` is ______. 

Options

• (∼ p) ∧ q

• (p) ∧ (∼ q)

• (∼ p) ∨ (∼ q)

• (∼ p) ∧ (∼ q)

Answer 1

Negation of the Boolean expression `p Leftrightarrow (q \implies p)` is (∼ p) ∧ (∼ q).

Explanation:

Given expression is `p Leftrightarrow (q \implies p)`

`∼ (p leftrightarrow (q rightarrow p))`

`∼ (p leftrightarrow q)` = (p∧ ∼ q) ∨ (q∧ ∼ p)

`∼ (p leftrightarrow (q rightarrow p))` = (p∧ ∼ (q `rightarrow` p)) ∨ ((q `rightarrow` p)∧ ∼ p)

`(p∧ ∼ (q rightarrow p))` = p ∧ (q∧ ∼ p) = (p∧ ∼ p) ∧ q = c

`(q rightarrow p) ∧ ∼ p` = (∼ q ∨ p) ∧∼ p = ∼ p ∧ (∼ q ∨ p)

= (∼ p∧ ∼ q) ∨ (∼ p ∧ p) = ∼ p∧ ∼ q

∼ `(p leftrightarrow (q rightarrow p))` = c ∨ (∼ p∧ ∼ q) = ∼ p∧ ∼ q