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  1. GATE CS
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  3. Discrete Mathematics

Which of the following is/are a tautology? A. a or b -> b and c B. a and b -> b or c C. a or b -> (b -> c) D. a -> b -> (b -> c)

GATE 1992 · Discrete Mathematics · Propositional Logic · easy

Answer: Only B (a and b -> b or c) is a tautology, so the answer is B.

  1. B is a tautology: If a and b is True then b is True, hence b or c is True. The implication never has a True antecedent with a False consequent, so B is always True - a tautology.
  2. A is falsifiable: Take a=True, b=False, c=False. Then a or b = True but b and c = False, giving True -> False = False. So A is not a tautology.
  3. C is falsifiable: Take a=False, b=True, c=False. Then a or b = True and b -> c = True -> False = False, so the whole formula is True -> False = False. C is not a tautology.
  4. D is falsifiable: With right-association, take a=True, b=True, c=False. Then b -> c = False, so b -> (b -> c) = True -> False = False, and a -> False = True -> False = False. D is not a tautology.