Question

Which of the following statements is a tautology?

• A. \(((∼p)∨q)⇒p\)
• B. \(p⇒((∼p)∨q)\)
• C. \(((∼p)∨q)⇒q\)
• D. \(q⇒((∼p)∨q)\)

Answer 1

The Correct Option is D

To determine which of the given statements is a tautology, we need to analyze each option using truth tables or logical reasoning.

A tautology is a statement that is always true, regardless of the truth values of its components. We will check each logical expression to see if it satisfies this condition.

1. (((∼p)∨q)⇒p)

   This expression is: "If (not p or q), then p". The expression is true if whenever the premise (∼p ∨ q) is true, the conclusion p is also true. However, consider the case where p = \text{false} and q = \text{true}, then (∼p ∨ q) is true but p is false. So, this is not a tautology.

2. (p⇒((∼p)∨q))

   This expression is: "If p, then (not p or q)". The expression is true if the premise p is false or the conclusion (∼p ∨ q) is true. If p = \text{true} and q = \text{false}, then (∼p ∨ q) is false. So, this is not a tautology.

3. (((∼p)∨q)⇒q)

   This expression is: "If (not p or q), then q". The expression is only false when the premise (∼p ∨ q) is true and the conclusion q is false. If p = \text{true} and q = \text{false}, then (∼p ∨ q) is true but q is false. So, this is not a tautology.

4. (q⇒((∼p)∨q))

   This expression is: "If q, then (not p or q)". The expression is a tautology because whenever q is true, (∼p ∨ q) is guaranteed to be true because q itself is a disjunct that makes the entire disjunction (∼p ∨ q) true. Thus, it is impossible for this implication to be false. Hence, this is a tautology.

Therefore, the correct statement that is a tautology is option 4: q⇒((∼p)∨q)