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Question
With proper justification, state the negation of the following.
(p → q) ∨ (p → r)
Solution
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
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