Question

Let r ∈ {p, q, ~p, ~q} be such that the logical statement r ∨ (~p) ⇒ (p ∧ q) ∨ r is a tautology. Then r is equal to :

• A. p
• B. q
• C. ~p
• D. ~q

Answer 1

The Correct Option is C

To determine the value of \( r \) that makes the logical statement \( r \lor (\sim p) \Rightarrow (p \land q) \lor r \) a tautology, we need to analyze the given expression.

Step 1: Understanding the Implication

The logical expression given is an implication: \( r \lor (\sim p) \Rightarrow (p \land q) \lor r \).

An implication \( A \Rightarrow B \) is a tautology if and only if \( A \lor \sim B \) is always true.

Step 2: Expressing the Implication as a Disjunction

Rewrite the implication using the equivalent disjunction form:

r \lor (\sim p) \lor \sim((p \land q) \lor r)

Apply De Morgan's laws to the negation:

r \lor (\sim p) \lor ((\sim p) \lor (\sim q) \land (\sim r))

Step 3: Simplifying the Expression

Simplify the statement by distributing \( r \) and \( \sim p \):

r \lor (\sim p) \lor ((\sim p) \land (\sim q) \land (\sim r))

This simplifies further to:

• r
• r \land (\sim r) = 0 (contradiction; omitted)
• r \land (\sim q)
• r \land (\sim p) if \( \sim q \) held.
• (\sim p)\, holds; covers potential invalid states.

Step 4: Evaluating for Tautology Condition

For the entire expression to be always true (a tautology), \(\sim p\) must hold.

None of \( p \) or \( q \) alone permit a failsafe tautology without conflict.

Conclusion: The value of \( r \) that makes the statement a tautology is \sim p.