Question

Which of the following Boolean expression is a tautology?

• A. \((p\wedge q)\vee(p\to q)\)
• B. \((p\wedge q)\vee(p\vee q)\)
• C. \((p\wedge q)\to(p\to q)\)
• D. \((p\wedge q)\wedge(p\to q)\)

Hint:

Look for a truth assignment that makes an expression false; \((p\wedge q)\to(p\to q)\) can never be false because \(p\wedge q\) forces \(q\) to be true.

Answer 1

The Correct Option is C

Test with truth values.
(1) \(p=T,q=F\): \((F)\vee(F)=F\), not a tautology.
(2) \(p=F,q=F\): \((F)\vee(F)=F\), not a tautology.
(3) \((p\wedge q)\to(p\to q)\) is false only if \(p\wedge q\) is true and \(p\to q\) is false; but \(p\wedge q\) true means \(p=q=T\), making \(p\to q\) true. So it is never false: tautology.
(4) \(p=F\): \((F)\wedge(T)=F\), not a tautology.
Answer: (3) \((p\wedge q)\to(p\to q)\)