Problem 22 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(p \rightarrow(\sim p \vee \sim q)\) is false, then the truth values of \(p\) and \(q\) are respectively. (a) \(\mathrm{T}_{2} \mathrm{~F}\) (b) \(\mathrm{F}, \mathrm{F}\) (c) \(\mathrm{F}_{2} \mathrm{~T}\) (d) \(\mathrm{T}, \mathrm{T}\)

Step-by-Step Solution

Answer

The correct answer is option (d) \(\mathrm{T}, \mathrm{T}\).

Step 1: Analyze the Implication

The logical statement is an implication: \( p \rightarrow (\sim p \vee \sim q) \). An implication is false only when the antecedent \(p\) is true and the consequent \((\sim p \vee \sim q)\) is false.

Step 2: Explore the Consequent

The consequent is a disjunction: \( \sim p \vee \sim q \). This statement is false only when both \(\sim p\) and \(\sim q\) are false. For \(\sim p\) to be false, \(p\) must be true, and for \(\sim q\) to be false, \(q\) must be true.

Step 3: Determine Truth Values

To make \( p \rightarrow (\sim p \vee \sim q) \) false, both conditions from the previous steps must hold: \( p \) is true and \( q \) is true. This makes the implication false, as required.

Step 4: Compare with Options

The truth values of \(p\) and \(q\) being both true corresponds to option (d) \(\mathrm{T}, \mathrm{T}\).

Key Concepts

Implication

In mathematical logic, an implication is a statement formed by two propositions, where one proposition leads to the logical outcome of the other. The typical form of an implication is represented as \( p \rightarrow q \). The proposition before the arrow \( p \) is known as the antecedent, while the one following the arrow \( q \) is the consequent.

An implication \( p \rightarrow q \) can be interpreted as "if \( p \), then \( q \)". Importantly, this type of logical statement is only false in one scenario: when the antecedent \( p \) is true, but the consequent \( q \) is false. In all other cases—whether both propositions are true, both are false, or the antecedent is false—the implication holds True.

Understanding implications is crucial in logic, as they help us to deduce conclusions based on certain premises. It's a foundational tool for argumentation and building logical expressions.

Truth Values

Truth values are the building blocks of logical expressions. In basic logic, any proposition or logical statement can have one of two truth values: True (T) or False (F). These values help determine the outcome of logical operations and expressions.

- **True (T):** The statement accurately describes a fact or a situation that holds true.- **False (F):** The statement inaccurately describes a fact or a situation and does not hold true.

In the context of implications, understanding the truth values of components like the antecedent and consequent is key to determining the truth value of the entire logical expression. For example, in the implication \( p \rightarrow q \), if \( p \) is true and \( q \) is false, then the whole implication is false. However, if both are true, or \( p \) is false, the implication is generally considered true.

Disjunction

A disjunction is another fundamental concept in logical operations. It is represented by the symbol \( \vee \), and it refers to the logical "or." A disjunction \( p \vee q \) is true when at least one of the propositions \( p \) or \( q \) is true.

Understanding disjunctions is particularly useful when analyzing compound logical statements. In our exercise, the disjunction \( \sim p \vee \sim q \) means that the overall expression will be true if either \( \sim p \) is true or \( \sim q \) is true, or both. Here's a breakdown:

• If \( \sim p \) is true, then \( p \) is false.
• If \( \sim q \) is true, then \( q \) is false.
• The expression is false only if both \( p \) and \( q \) are true, making both \( \sim p \) and \( \sim q \) false.

Understanding how disjunction operates is key in handling the logical operators and crafting complex logical statements.

Logical Operators

In the realm of mathematical logic, logical operators are the symbols or words used to connect statements or propositions, forming complex logical expressions. There are several logical operators, each with unique functionalities.

Here are a few essential operators:

• **Negation (\( \sim \)):** This operator inverts the truth value of a proposition. For example, if \( p \) is true, \( \sim p \) is false, and vice versa.
• **Conjunction (\( \wedge \)):** A conjunction \( p \wedge q \) is true only when both propositions \( p \) and \( q \) are true.
• **Disjunction (\( \vee \)):** As explained, \( p \vee q \) is true if either \( p \) or \( q \) or both are true.
• **Implication (\( \rightarrow \)):** As discussed, this is true except when a true antecedent leads to a false consequent.

Logical operators form the syntax of logically expressing real-world situations and reasoning through logical arguments. By combining propositions with these operators, we can evaluate the truthfulness of more complex statements.