Problem 10 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the truth value of the statement \(p \rightarrow(\sim q \vee r)\) is false (F), then the truth values of the statements \(p, q, r\) are respectively. \(\quad\) (a) \(\mathrm{T}, \mathrm{T}, \mathrm{F}\) (b) \(\mathrm{T}, \mathrm{F}, \mathrm{F}\) (c) \(\mathrm{T}, \mathrm{F}, \mathrm{T}\) (d) \(\mathrm{F}, \mathrm{T}, \mathrm{T}\)

Step-by-Step Solution

Answer

The truth values are T, T, F (option a).

Step 1: Understand the Conditional Statement

The conditional statement is \(p \rightarrow (\sim q \vee r)\). This means "if \(p\) then \((\sim q \vee r)\)." For a conditional statement to be false, the antecedent \(p\) must be true, and the consequent \((\sim q \vee r)\) must be false.

Step 2: Analyze the Consequent for Falsehood

For \((\sim q \vee r)\) to be false, the component \(\sim q\) must be false, and \(r\) must be false. This is because a disjunction is false only if both components are false.

Step 3: Check Negation of q

Since \(\sim q\) is false, it means \(q\) must be true because the negation operator (\(\sim\)) makes true values false and vice versa.

Step 4: Compile Possible Truth Values

From the above reasoning, we find that for the entire expression \(p \rightarrow (\sim q \vee r)\) to be false: \(p\) must be true, \(q\) must be true, and \(r\) must be false.

Step 5: Select the Correct Answer

The correct set of truth values satisfying the condition where \(p \rightarrow (\sim q \vee r)\) is false, is \(\mathrm{T}, \mathrm{T}, \mathrm{F}\). Therefore, the correct answer is option (a).

Key Concepts

Conditional Statements

In propositional logic, conditional statements form a fundamental part of logical reasoning. A conditional statement is expressed as "if...then..." and is written mathematically using the arrow symbol: \( p \rightarrow q \). This expression means "if \( p \), then \( q \)." The part before the arrow "\( p \)" is known as the antecedent, while the part after the arrow "\( q \)" is the consequent.
If you want to determine the truth value of a conditional statement, there’s one primary rule to keep in mind:

• The statement \( p \rightarrow q \) is false only if \( p \) is true and \( q \) is false.

A conditional statement prompts you to consider scenarios based on this rule. In our specific problem, you need to examine what conditions make the statement \( p \rightarrow (\sim q \vee r) \) false, implying \( p \) must be true and \((\sim q \vee r)\) must be false.

Truth Tables

Truth tables are handy tools used to determine the validity and truth value of logical statements. They help illustrate how different combinations of truth values for given propositions affect the truth value of the overall statement.
A truth table lists every possible combination of truth values for the involved propositions and shows the result for each configuration. Consider a disjunction \(\sim q \vee r\):

• If \(\sim q\) is true and \( r \) is true, the disjunction is true.
• If \(\sim q\) is true and \( r \) is false, the disjunction is still true.
• If \(\sim q\) is false and \( r \) is true, the disjunction remains true.
• Only if both \(\sim q\) and \( r \) are false does the disjunction become false.

This understanding helps to decipher how each component contributes to the logical statement's overall truth in exercises like the given one.

Logical Operators

Logical operators are symbols or words used to connect propositions to form compound statements. In our exercise about propositional logic, we are dealing with the operators implication (\(\rightarrow\)), negation (\(\sim\)), and disjunction (\(\vee\)). These operators have specific meanings and behaviors:

• The implication (\(\rightarrow\)) operator indicates a conditional relationship between two propositions. Recall it is only false when the first proposition is true and the second is false.
• The negation (\(\sim\)) operator flips the truth value of a proposition; if \(q\) is true, \(\sim q\) is false, and vice versa.
• The disjunction (\(\vee\)) operator provides an "or" relationship. It would be true if at least one of the components (such as \(\sim q\) or \(r\)) is true, and only false when both components are false.

Understanding how these logical operators work together helps to determine the truth or falsehood of complex statements, enabling you to solve exercises that ask for specific truth values.