Statement $(p\vee q)\vee r\Leftrightarrow p\vee(q\vee r)$ is:

Question

Accepted Answer

The statement $(p \vee q) \vee r \Leftrightarrow p \vee (q \vee r)$ is a logical expression that involves the associative property of the logical OR operator ($\vee$). The associative property states that the way in which operands are grouped in an expression does not affect the result, as long as the operations are the same. In this case, the expression is dealing with the OR operation, which is associative.

1. **Explanation for Option 1 (Tautology):**
   - A tautology is a statement that is always true, regardless of the truth values of the individual variables involved. The expression $(p \vee q) \vee r \Leftrightarrow p \vee (q \vee r)$ is a tautology because the associative property of the OR operation ensures that the grouping of the variables does not affect the outcome. No matter what truth values $p$, $q$, and $r$ take (true or false), the expression will always evaluate to true. This is because both sides of the equivalence will yield the same result for any combination of truth values, confirming that the expression is indeed a tautology.

2. **Explanation for Option 2 (Fallacy):**
   - A fallacy is a statement that is always false. The given expression is not a fallacy because, as explained, it is always true due to the associative property of the OR operation. Therefore, Option 2 is incorrect.

3. **Explanation for Option 3 (Neither tautology nor fallacy):**
   - This option suggests that the expression is neither always true nor always false. However, as established, the expression is always true due to the associative property, making it a tautology. Therefore, Option 3 is incorrect.

4. **Explanation for Option 4 (Both tautology and fallacy):**
   - It is logically impossible for a statement to be both a tautology and a fallacy simultaneously. A tautology is always true, while a fallacy is always false. Since the expression is always true, it cannot be a fallacy. Therefore, Option 4 is incorrect.

In conclusion, the expression $(p \vee q) \vee r \Leftrightarrow p \vee (q \vee r)$ is a tautology because it is always true due to the associative property of the OR operation. This supports Option 1 as the correct answer.