Question

Equivalent statement to (p\(\to\)q) \(\vee\) (r\(\to\)q) will be

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

Answer 1

The Correct Option is A

To find an equivalent statement for \((p \to q) \vee (r \to q)\), we need to analyze the logical structures and apply equivalent transformations.

1. The conditional implication \(p \to q\) can be rewritten using logical equivalence as \(\neg p \vee q\).
2. Similarly, the implication \(r \to q\) can be rewritten as \(\neg r \vee q\).
3. Now the given expression \((p \to q) \vee (r \to q)\) can be transformed using the above transformations:
   • \((p \to q) \vee (r \to q) = (\neg p \vee q) \vee (\neg r \vee q)\)
4. Apply the Distributive Law in logic, which states \(A \vee (B \vee C) \equiv (A \vee B) \vee (A \vee C)\):
   • \((\neg p \vee q) \vee (\neg r \vee q) = \neg p \vee (q \vee \neg r) \vee q\)
5. Since \(q \vee q = q\), we simplify this to:
   • \(\neg p \vee q\)
6. We can now convert this back to its conditional form:
   • \(\neg p \vee q \equiv p \to q\)
7. By applying resolution principle, \((p \vee r) \to q\) is equivalent to our transformed expression.

The correct answer from the given options is (p ∧ r) → q, which encompasses cases where both p and r being true implies q.