Question

For p and q, consider: (a) (~ q ∧ (p → q)) → ~ p, (b) ((p ∨ q) ∧ ~ p) → q. Which is correct ?

• (a) is a tautology but not (b).
• (b) is a tautology but not (a).
• (a) and (b) both are tautologies.
• (a) and (b) both are not tautologies.

Hint:

The logical form "If (A and B) then A" is always a tautology. Use distribution laws to simplify the antecedent.

Answer 1

The Correct Option is C

To determine if the given propositions are tautologies, we need to analyze each of them using logical operations. A tautology is a statement that is true in every possible interpretation.

Consider statement (a): \((~ q \land (p \to q)) \to ~ p\). 

• The implication \(p \to q\) can be rewritten as \(~p \lor q\).
• So, \((~ q \land (p \to q))\) becomes \((~q \land (~p \lor q))\).
• This simplifies to \((~q \land ~p) \lor (~q \land q)\).
• Since \(( ~q \land q )\) is always false, the expression reduces to \((~q \land ~p)\).
• The implication \((~q \land ~p) \to ~p\) is true because if \(~q\) and \(~p\) are true, then \(~p\) is true.
• Thus, statement (a) is a tautology.

Consider statement (b): \(((p \lor q) \land ~ p) \to q\).

• The conjunction \(((p \lor q) \land ~ p)\) is applied as follows:
  • For the statement to hold true, at least one of \(p\) or \(q\) must be true, but \(p\) must be false.
  • So, \(p\) is false, then \(q\) must be true for the implication to hold:
  • Thus, under all conditions, if \(p \lor q\) is true and \(p\) is false, \(q\) must be true for this statement to be true.
• Therefore, statement (b) is a tautology.

Hence, both (a) and (b) are tautologies, which means the correct answer is:

(a) and (b) both are tautologies.