Question

If the Boolean expression \((p \wedge q) \odot (p \otimes q)\) is a tautology, then \(\odot\) and \(\otimes\) are respectively given by :

• A. \(\vee , \rightarrow\)
• B. \(\rightarrow , \rightarrow\)
• C. \(\wedge , \vee\)
• D. \(\wedge , \rightarrow\)

Hint:

Test each option: only the one where the whole expression is true for every truth assignment of p, q is a tautology.

Answer 1

The Correct Option is B

Option (1): \((p\wedge q)\vee(p\rightarrow q)\). For p = T, q = F both parts are F, so the expression is F. Not a tautology.
Option (2): \((p\wedge q)\rightarrow(p\rightarrow q)\). If \(p\wedge q\) is true then p, q are both T and \(p\rightarrow q\) is T; if \(p\wedge q\) is false the implication is T. Always true: a tautology.
Option (3): \((p\wedge q)\wedge(p\vee q)\) is equivalent to \(p\wedge q\), false when p = F. Not a tautology.
Option (4): \((p\wedge q)\wedge(p\rightarrow q)\) is equivalent to \(p\wedge q\), false when p = F. Not a tautology.
Answer: (2) \(\rightarrow , \rightarrow\)