Question

Which of the following statements is a tautology?

• A. $((\sim p) \vee q) \Rightarrow p$
• B. $p \Rightarrow((\sim p ) \vee q )$
• C. $((\sim p ) \vee q ) \Rightarrow q$
• D. \(q \Rightarrow((\sim p) \vee q)\)

Answer 1

The Correct Option is D

To determine which of the given statements is a tautology, we need to evaluate each statement logically. A tautology is a statement that is true under all possible interpretations or truth values of its components.

Evaluating Each Statement

1. Consider the statement \( ((\sim p) \vee q) \Rightarrow p \).
   • Let's evaluate this: The expression \( ((\sim p) \vee q) \) is true if either \( \sim p \) is true or \( q \) is true.
   • The implication \( ((\sim p) \vee q) \Rightarrow p \) is true if \( p \) is true or if \( ((\sim p) \vee q) \) is false.
   • To check tautology, it must be true for all \( p \) and \( q \). If we assign \( p = \text{False} \) and \( q = \text{False} \), \((\sim p) \vee q\) becomes True, and the implication becomes False. Thus, it's not a tautology.
2. Consider the statement \( p \Rightarrow((\sim p ) \vee q ) \).
   • The implication \( p \Rightarrow ((\sim p) \vee q) \) is true if \( (\sim p) \vee q \) is true whenever \( p \) is true.
   • For any true \( p \), the right side \( ((\sim p) \vee q) \) may not always be true; for example, \( p = \text{True} \), \( q = \text{False} \), results in False. Not a tautology.
3. Consider the statement \( ((\sim p) \vee q) \Rightarrow q \).
   • This statement implies that if \( ((\sim p) \vee q) \) is true, then \( q \) must also be true for the implication.
   • If \( q \) is False, but \( \sim p \) is True, the implication is false. Thus, not a tautology.
4. Consider the statement \( q \Rightarrow ((\sim p) \vee q) \).
   • For this implication \( q \Rightarrow ((\sim p) \vee q) \) to be true, whenever \( q \) is true, \( ((\sim p) \vee q) \) must also be true.
   • Since \( (\sim p) \vee q \) is certainly true when \( q \) is true, this implication holds universally.
   • Thus, this statement is a tautology because it is true for all values of \( p \) and \( q \).

From the above evaluations, the statement \( q \Rightarrow ((\sim p) \vee q) \) is a tautology because it is always true regardless of the truth values of \( p \) and \( q \).