Let a, b, c, and d be propositions. Assume that the equivalences a ↔ (b ∨ ¬b)…
2000
Let a, b, c, and d be propositions. Assume that the equivalences a ↔ (b ∨ ¬b) and b ↔ c hold. Then the truth value of the formula ((a ∧ b) → ((a ∧ c) ∨ d)) is always:
- A.
True
- B.
False
- C.
Same as the truth value of b
- D.
Same as the truth value of d
Attempted by 20 students.
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Correct answer: A
The expression b ∨ ¬b is a tautology, so a ↔ (b ∨ ¬b) forces a to be true.
The condition b ↔ c means b and c always have the same truth value.
Now simplify the formula: ((a ∧ b) → ((a ∧ c) ∨ d)). Since a is true, this becomes b → (c ∨ d). Since c has the same truth value as b, it becomes b → (b ∨ d).
If b is true, then b ∨ d is true. If b is false, the implication is true vacuously. Therefore the formula is always true.