Question

With proper justification, state the negation of the following.

(p → q) ∨ (p → r)

Answer 1

Step 1: Expressing Implications in Terms of Logical Operators

p → q ≡ ∼p ∨ q

p → r ≡ ∼p ∨ r

(p → q) ∨ (p → r)

(∼p ∨ q) ∨ (∼p ∨ r)

Using the associative and distributive properties of logical operators:

∼p ∨ (q ∨ r)

Step 2: Negation of the Statement

∼[∼p ∨ (q ∨ r)]

Using De Morgan’s Theorem:

∼(∼p) ∧ ∼(q ∨ r)

p ∧ (∼q ∧ ∼r)

p ∧ ∼q ∧ ∼r

Step 3: Interpretation

The negation of the given statement means:

• p is true.
• q is false.
• r is false.

Thus, the negation of (p → q) ∨ (p → r) is:

p ∧ ∼ q ∧ ∼r