Problem 33 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(p\) and \(q\) be any two logical statements and \(\mathrm{r}: \mathrm{p} \rightarrow(-p \vee q) .\) If \(r\) has a truth value \(F\), then the truth values of \(p\) and \(q\) are respectively (a) \(\mathrm{F}_{2} \mathrm{~F}\) (b) \(\mathrm{T}_{2} \mathrm{~T}\) (c) \(\mathrm{T}_{2} \mathrm{~F}\) (d) \(\mathrm{F}, \mathrm{T}\)

Step-by-Step Solution

Answer

The truth values of \( p \) and \( q \) are True and False, respectively.

Step 1: Understand the Conditional Statement

We are given the statement \( r: p \rightarrow (-p \vee q) \). This is a conditional statement, where \( p \rightarrow (-p \vee q) \) means 'if \( p \) then \( -p \vee q \).' For a conditional statement \( A \rightarrow B \) to be false, \( A \) must be true and \( B \) must be false.

Step 2: Analyze Conditions for Falsehood

For the statement \( p \rightarrow (-p \vee q) \) to be false, \( p \) must be true, and \( -p \vee q \) must be false. That means both \( -p \) and \( q \) have to be false simultaneously, giving us \( -p = \text{False} \Rightarrow p = \text{True} \) and \( q = \text{False} \).

Step 3: Determine the Values of p and q

From Step 2, we found that \( p = \text{True} \) and \( q = \text{False} \) satisfy the condition where the statement \( p \rightarrow (-p \vee q) \) is false.

Key Concepts

Logical Statements

Logical statements are declarative sentences that can either be true or false, but not both. They are the building blocks of logic used extensively in mathematics, computer science, and philosophy. Logical statements are often denoted by variables such as \(p\), \(q\), and \(r\) to represent different propositions. For example:

• \(p\): It is raining.
• \(q\): I will take an umbrella.
• \(r\): \(p \rightarrow (-p \vee q)\) means 'If \(p\), then \(-p \vee q\)'.

Understanding logical statements is crucial as they form the basis of constructing logical expressions and solving logical problems in fields like mathematics and computer programming.

Truth Values

In logic, each statement is assigned a truth value, which is either True (T) or False (F). Truth values help evaluate the validity of logical expressions. They are essential in building truth tables, which are used to determine the truthfulness of compound statements.Consider the statement \(p \rightarrow q\):

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

Thus, understanding and applying truth values is imperative to evaluate whether complex logical statements hold true or false under different conditions.

Conditional Statements

Conditional statements, also known as implications, have the form \(A \rightarrow B\). They are read as 'if \(A\), then \(B\)'. Such statements play a pivotal role in logical reasoning and problem-solving. The truth value of a conditional statement depends on its components:- If \(A\) is true and \(B\) is true, \(A \rightarrow B\) is true.- If \(A\) is true and \(B\) is false, \(A \rightarrow B\) is false.- If \(A\) is false, \(A \rightarrow B\) is true, regardless of \(B\).This characteristic makes conditional statements unique as the implication is only false when the first part (antecedent) is true and the second part (consequent) is false—an essential concept in deductive reasoning.

Negation

Negation is a logical operation that takes a statement and reverses its truth value. It is represented by the symbol \(-\) or 'not'. The negation of a statement \(p\) is expressed as \(-p\), interpreted as 'not \(p\)'. The effects of negation are straightforward:

• If \(p\) is true, \(-p\) is false.
• If \(p\) is false, \(-p\) is true.

Negation is a fundamental element in constructing logical expressions and helps in forming the negative or opposite of any statement. In the context of conditional statements, understanding the negation of variables like \(p\) aids in analyzing the truth conditions of the expressions.

Logical Disjunction

Logical disjunction refers to the operation performed when using the "or" operator, denoted as \(\vee\). In logic, the statement \(A \vee B\) evaluates to true if at least one of the individual statements \(A\) or \(B\) is true. Here are the truth conditions:

• If \(A\) is true, \(A \vee B\) is true, regardless of \(B\).
• If \(B\) is true, \(A \vee B\) is true, regardless of \(A\).
• If both \(A\) and \(B\) are false, \(A \vee B\) is false.

Logical disjunction captures the essence of inclusive "or", and it is crucial in scenarios where multiple conditions can satisfy an outcome. It is widely used in decision-making processes and programming.