Without using truth table, show that (p ∨ q) → r ≡ (p → r) ∧ (q → r)

EXERCISE 1.9Q 1.5   PAGE 22

Exercise 1.9 | Q 1.5 | Page 22

Without using truth table, show that

(p ∨ q) → r ≡ (p → r) ∧ (q → r)

SOLUTION

L.H.S.

≡ (p ∨ q) → r

≡ ~ (p ∨ q) ∨ r          ....[p → q → ~ p ∨ q]

≡ (~ p ∧ ~ q) ∨ r       ....[De Morgan’s law] 

≡ (~ p ∨ r) ∧ (~ q ∨ r)       .....[Distributive law]

≡ (p → r) ∧ (q → r)             .....[p → q → ~ p ∨ q]

= R.H.S.