Problem 13 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

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

Step-by-Step Solution

Answer

Options (C) and (D) make the statement false.

Step 1: Understand 'if and only if' (IFF)

The logical statement 'if and only if', denoted as \( p \iff q \), means that \( p \) is true if and only if \( q \) is true. In other terms, both \( p \) and \( q \) must have the same truth value for \( p \iff q \) to be true.

Step 2: Analyze the truth table for IFF

To uncover the truth values of \( p \iff q \), construct its truth table: - If \( p \) and \( q \) both are true, then \( p \iff q \) is true.- If \( p \) is true and \( q \) is false, then \( p \iff q \) is false.- If \( p \) is false and \( q \) is true, then \( p \iff q \) is false.- If \( p \) and \( q \) both are false, then \( p \iff q \) is true.

Step 3: Determine when IFF is false

From the truth table, \( p \iff q \) is false when \( p \) and \( q \) are of differing truth values: either \( p \) is true and \( q \) is false, or \( p \) is false and \( q \) is true.

Step 4: Apply to given options and conclude

Check each option:- (A) \( p \) is true, \( q \) is true. \( p \iff q \) is true.- (B) \( p \) is false, \( q \) is false. \( p \iff q \) is true.- (C) \( p \) is true, \( q \) is false. \( p \iff q \) is false.- (D) \( p \) is false, \( q \) is true. \( p \iff q \) is false.Hence, options (C) and (D) make the statement false.

Key Concepts

Truth Table

A truth table is an essential tool in understanding logical statements. It lists all possible combinations of truth values for a set of propositions and shows the corresponding result of a logical expression. Each row of the table represents a different scenario of truth values for the propositions involved. In our case with the statement "if and only if" (IFF), the propositions are denoted as \( p \) and \( q \).

Let's simplify with a truth table for \( p \iff q \):

• If both \( p \) and \( q \) are true, the outcome is true.
• If \( p \) is true and \( q \) is false, the outcome is false.
• If \( p \) is false and \( q \) is true, the outcome is false.
• If both \( p \) and \( q \) are false, the outcome is true.

This method provides a clear visual representation that helps in analyzing logical statements effectively.

If and Only If (IFF)

The logical connector "if and only if" is symbolized as \( p \iff q \). This means that both components \( p \) and \( q \) must share the same truth value for the statement to be true. Essentially, \( p \) being true must imply \( q \) is true, and vice versa. If one is true and the other is false, then the statement is false.

This is known as a biconditional statement, as it sets a double implication. Therefore, it emphasizes that the truth of one side entirely depends on the truth of the other. This unique property differentiates it from other logical connectors like "and" or "or", and it is crucial in determining under which conditions complex statements maintain their truth status.

Logical Statement Analysis

To conduct a logical statement analysis, we explore how individual propositions combine to form a complex statement. By breaking down these elements, we observe each interaction, aiming to determine whether the overall statement is true or false.

In the context of \( p \iff q \), our task is to scrutinize the relationship between \( p \) and \( q \). Analyzing the conditions under which these propositions lead to a true or false outcome helps college or school students work through contextual logic problems efficiently.

Proper logical analysis involves understanding the rules behind logical operations and applying them to see the bigger picture. Comprehending compound statements through this method ensures students can not only grasp basic logic but also apply it in diverse scenarios.

Conditional Statements

Conditional statements are fundamental in logical reasoning and are commonly represented in the form of "If...then..." They describe a scenario where the truth of one proposition necessitates the truth of another. However, this differs slightly from the "if and only if" condition, which requires both propositions to share identical truth values.

In typical conditional statements, symbolized by \( p \Rightarrow q \), if \( p \) (the antecedent) is true, then \( q \) (the consequent) must also be true. Otherwise, the statement can hold its truth value in other conditions without requiring both propositions to be identical.

This fundamental form of logic forms the building block of understanding more complex logical structures, allowing one to deduce consequences from given conditions. Mastery of conditional logic is pivotal to understanding more intricate logical equivalences and the broader applications in mathematics, computer science, and philosophy.