Problem 11 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Truth value of the statement "if \(p\) then \(q\) " is false when (A) \(p\) is true, \(q\) is true (B) \(p\) is true, \(q\) is false (C) \(p\) is false, \(q\) is true (D) \(p\) is false, \(q\) is false

Step-by-Step Solution

Answer

The statement is false for option (B): \( p \) is true, \( q \) is false.

Step 1: Understand Conditional Statements

An 'if-then' statement "if \( p \) then \( q \)" is a logical statement that is only false when the antecedent \( p \) is true and the consequent \( q \) is false. This is the scenario where the promise of the conditional is broken.

Step 2: Evaluate Option A

In option (A), \( p \) is true and \( q \) is true. In this case, the statement is true because both \( p \) and \( q \) are satisfied.

Step 3: Evaluate Option B

In option (B), \( p \) is true and \( q \) is false. This is the only scenario where the statement "if \( p \) then \( q \)" is false, as \( p \) being true does not lead to \( q \) being true.

Step 4: Evaluate Option C

In option (C), \( p \) is false and \( q \) is true. The statement would be true in this case because it does not matter what \( q \) is if \( p \) is false; the conditional is vacuously true.

Step 5: Evaluate Option D

In option (D), both \( p \) and \( q \) are false. The statement is true because if \( p \) is false, the truth value of \( q \) doesn't affect the overall truth of the conditional statement.

Key Concepts

Understanding Logic in Conditional Statements

Logic is the foundation of reasoning, helping us to draw conclusions based on given premises. In the world of logic, a common type of statement is known as the **conditional statement**. It's formed using an "if-then" structure, like "if \( p \) then \( q \)." Here, \( p \) represents the 'antecedent' or the first part, and \( q \) is the 'consequent,' the second part.
A conditional statement is concerned not only with whether \( p \) or \( q \) are true or false on their own, but importantly with their relationship. It essentially says: if the condition \( p \) is met, then we can expect condition \( q \) to follow.

• For instance, think of the statement "if it rains \( p \), then the ground will be wet \( q \)." Even if you never actually check if the ground is wet, as long as it rains, the conditional statement holds true unless the ground isn't wet.
• It's vital to remember that "if \( p \) then \( q \)" is only false when \( p \) is true and \( q \) is not, breaking the expected outcome.

Decoding Truth Tables

Truth tables are a useful tool that help visualize how the truth values of components affect the overall truth of the logical statement. Consider the truth table for "if \( p \) then \( q \)":

• If \( p \) is true and \( q \) is true, the statement is true since the promise specified by the conditional is fulfilled.
• If \( p \) is true and \( q \) is false, the statement is false because the expected outcome \( q \) does not occur, violating the logical condition.
• If \( p \) is false and \( q \) is true, the statement remains true, since the condition "\( p \)" does not apply. Such cases are called vacuously true.
• Finally, if \( p \) is false and \( q \) is false, the statement is still true for the same reason as before; \( p \) being false makes "if \( p \) then \( q \)" vacuously true.

Truth tables are essentially a map that leads us to determine exactly when complex statements are true or false. They simplify analyzing logical implications.

Antecedent and Consequent in Conditional Statements

In a conditional statement, the terms 'antecedent' and 'consequent' support the structure of 'if-then' logic.

• **Antecedent**: This is the 'if' part of the statement, marking the premise or hypothesis. In a formula like "if \( p \) then \( q \)", \( p \) is the antecedent. It's what you're assuming or starting from.
• **Consequent**: This is the 'then' part, representing the conclusion or result. In our example, \( q \) is the consequent, representing what happens when the antecedent \( p \) is true.

Understanding these terms is key to unpacking how conditional logic works. The antecedent sets the stage, and the consequent plays out upon the satisfaction of the antecedent. In terms of truth value, the transition from antecedent to consequent is what we scrutinize using logic and truth tables.