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Question
Using truth table, prove that the statement patterns p ↔ q and (p ∧ q) ∨ (~ p ∧ ~ q) are logically equivalent.
Sum
Solution
p ↔ q ≡ (p ∧ q) ∨ (~ p ∧ ~ q)
| I | II | III | IV | V | VI | VII | VIII |
| p | q | ~p |
~q |
p ↔ q | p ∧ q | ~ p ∧ ~ q | (p ∧ q) ∨ (~ p ∧ ~ q) |
| T | T | F | F | T | T | F | T |
| T | F | F | T | F | F | F | F |
| F | T | T | F | F | F | F | F |
| F | F | T | T | T | F | T | T |
Columns V and VIII are identical.
∴ p ↔ q ≡ (p ∧ q) ∨ (~ p ∧ ~ q)
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