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Question
Using the truth table prove the following logical equivalence.
(p ∨ q) → r ≡ (p → r) ∧ (q → r)
Solution
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| p | q | r | p ∨ q | (p ∨ q) → r | p → r | q → r | (p → r) ∧ (q → r) |
| T | T | T | T | T | T | T | T |
| T | T | F | T | F | F | F | F |
| T | F | T | T | T | T | T | T |
| T | F | F | T | F | F | T | F |
| F | T | T | T | T | T | T | T |
| F | T | F | T | F | T | F | F |
| F | F | T | F | T | T | T | T |
| F | F | F | F | T | T | T | T |
The entries in columns 5 and 8 are identical.
∴ (p ∨ q) → r ≡ (p → r) ∧ (q → r)
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