Problem 5 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(p, q, r\) be three statements such that the truth value of \((p \wedge q) \rightarrow(\sim q \vee r)\) is \(\mathrm{F}\). Then the truth values of \(p, q, r\) are respectively: (a) \(\mathrm{T}, \mathrm{F}, \mathrm{T}\) (b) \(\mathrm{T}, \mathrm{T}, \mathrm{T}\) (c) \(\mathrm{F}, \mathrm{T}, \mathrm{F}\) (d) \(\mathrm{T}, \mathrm{T}, \mathrm{F}\)

Step-by-Step Solution

Answer

The truth values are \\( p = \mathrm{T}, q = \mathrm{T}, r = \mathrm{F} \\\), corresponding to option (d).

Step 1: Understanding the Conditional Statement

The given statement is \( (p \wedge q) \rightarrow (\sim q \vee r) \). A conditional statement \( A \rightarrow B \) is true except if A is true and B is false.

Step 2: Analyzing to Find When the Conditional is False

For the given statement to be false, \( p \wedge q\) must be true and \( \sim q \vee r \) must be false. Let's determine when each is true or false.

Step 3: Evaluating \\( p \wedge q\\\)

The expression \( p \wedge q \) is true only if both \( p \) and \( q \) are true.

Step 4: Evaluating \\( \sim q \vee r\\\)

The expression \( \sim q \vee r \) is false only if \(\sim q\) is false and \(r\) is false. This implies \(q\) must be true and \(r\) must be false.

Step 5: Determining Correct Truth Values

Since \( p \wedge q \) is true with \( q \mathrm{ = T}\ ext{ and }r = \mathrm{F}\), \( p \) must be true. Thus, \( p = \mathrm{T}, \), \( q = T \), \( r = F\).

Step 6: Identify the Correct Answer Choice

The truth values \(( p = \mathrm{T}, q = \mathrm{T}, r = \mathrm{F}) \) match answer choice (d).

Key Concepts

Logical Reasoning

Logical reasoning is all about using a structured way to come to a conclusion based on given information. In mathematical problems, statements are analyzed and evaluated for their truth values to make deductions. Logical reasoning is crucial in solving complex problems, especially in competitive exams like JEE Main Mathematics. By breaking down the statement logically, we can figure out which components must be true or false, as seen in the problem above.

• Logical reasoning often involves identifying premises and conclusions.
• It helps to convert statements into mathematical or logical expressions.
• This process requires understanding how different logical operations, like conjunctions and disjunctions, work.

In our exercise, logical reasoning is used to determine when a compound statement involving conjunction and disjunction becomes false, which helps identify the truth values of the individual statements.

Conditional Statements

Conditional statements, involving an "if-then" structure, are a key component of logical statements. The format such as \( A \rightarrow B \), asserts that if statement \( A \) is true, then \( B \) must be true as well for the whole statement to remain true. If \( A \) is true and \( B \) is false, then the entire conditional statement is false. This is the critical concept in our problem as the given conditional was false.
Here’s a simplified explanation:

• *Antecedent (A):* the initial part, "if part" (like \( p \wedge q \)).
• *Consequent (B):* the following part, "then part" (like \( \sim q \vee r \)).

Understanding this helps us know under what conditions a statement switches between true and false, as shown in our step-by-step solution where we identified the falsehood of \( \sim q \vee r \), leading to the determination of truth values.

Truth Values

Truth values are fundamental in logic, indicating whether a statement is correct (true) or incorrect (false). In logic, each statement can only be either true (T) or false (F). The truth value differs from subjective truths as it deals with logical, objective assessments. In our problem, determining whether the entire statement was false required evaluating each truth value of the components involved.
To determine truth values, consider:

• Applying logical operators like \( \wedge \), \( \vee \), and \( \rightarrow \).
• Knowing that \( p \wedge q \) is true only if both p and q are true.
• Understanding \( \sim q \vee r \) is false if both \( \sim q \) and \( r \) are false.

The approach is to deconstruct complex statements using these basic rules to see how they affect the truth value of the overall statement, helping us solve the exercise by knowing which statement conditions would make the compound statement false.