Question

The Truth Table Represents Statement P,Q,R

Answer 1

Recall that a ⇒ b ≡ ¬a ∨ b.

• r ⇒ (p ∧ q) ≡ ¬r ∨ (p ∧ q)

In row C, q is false so p ∧ q false, and r is true so ¬r is false.

¬r ∨ (p ∧ q) ≡ false ∨ false ≡ false

• r ⇒ (p ∨ q) ≡ ¬r ∨ (p ∨ q) = p ∨ q ∨ ¬r

In each of rows A, C, and E, at least one of p or q is true, so

p ∨ q ∨ ¬r = true

• (q ∧ r) ⇒ p ≡ ¬(q ∧ r) ∨ p ≡ (¬q ∨ ¬ r) ∨ p = p ∨ ¬q ∨ ¬r

In row E, p is false and both q and r are true, so ¬q and ¬r are both false.

false ∨ false ∨ false = false

• (q ∨ r) ⇒ p ≡ ¬(q ∨ r) ∨ p ≡ (¬q ∧ ¬r) ∨ p

In row E, p is false and both q and r are true, so both ¬q and ¬r are false.

(¬q ∧ ¬r) ∨ p ≡ (false ∧ false) ∨ false ≡ false ∨ false ≡ false