Problem 32 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Consider Statement-1: \((p \wedge \sim q) \wedge(\sim p \wedge q)\) is a fallacy. Statement- \(2:(p \rightarrow q) \leftrightarrow(\sim q \rightarrow \sim p)\) is a tautology. (a) Statement- 1 is true; Statement-2 is true; Statement-2 is a correct explanation for Statement-1. (b) Statement- 1 is true; Statement- 2 is true; Statement- 2 is not a correct explanation for Statement-1. (c) Statement- 1 is true; Statement- 2 is false. (d) Statement- 1 is false; Statement- 2 is true.

Step-by-Step Solution

Answer

(b) Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

Step 1: Analyze Statement 1

Statement 1 is \( (p \wedge \sim q) \wedge (\sim p \wedge q) \). This would require both \((p \wedge \sim q)\) and \( (\sim p \wedge q) \) to be true. However, it is not possible for both parts to be true simultaneously. When \(p = T \) and \(q = F \), \( \sim p = F \) and \( \sim q = T \). Conversely, when \(\sim p = T \) and \(\sim q = F \), \(p = F \) and \(q = T \), which contradicts each other. Therefore, \( (p \wedge \sim q) \wedge (\sim p \wedge q) \) is a fallacy.

Step 2: Analyze Statement 2

Statement 2 is \( (p \rightarrow q) \leftrightarrow (\sim q \rightarrow \sim p) \). This can be rewritten using logical equivalences. \((p \rightarrow q) \) is equivalent to \( ( \sim p \lor q) \) and \( (\sim q \rightarrow \sim p) \) is equivalent to \( (p \lor \sim q) \). The statement then becomes \( (\sim p \lor q) \leftrightarrow (p \lor \sim q) \). This is always true for all truth values of \( p \) and \( q \), making it a tautology.

Step 3: Evaluate Explanation Connection

Statement-1 discusses a logical fallacy which is a combination of propositions always resulting in falsehood. Statement-2's tautology is derived from logical equivalence transformations, not directly related to detecting fallacies. Thus, Statement-2's truth does not explain the fallacy described in Statement-1.

Key Concepts

Tautology

Tautology is an important concept in logic. It refers to a statement that is always true, regardless of the truth values of its individual components.

In logical terms, a tautology is a formula or assertion that is true in every possible interpretation. It will never result in a falsehood, no matter the input values of its variables.

For example, consider the logical statement:

• \( (p \rightarrow q) \leftrightarrow (\sim q \rightarrow \sim p) \)

This statement is a tautology. Why? Because for every true or false value of \( p \) and \( q \), the statement holds true. It’s like saying, "It will rain or it won't rain." Either way, the statement remains true.

Tautologies are helpful in logical reasoning and proofs. If we transform a complex problem into a tautology, we verify its truthfulness and avoid contradictions.

Logical Equivalence

Logical equivalence is a concept that shows how different statements can mean the same thing. Even if they look different, logically they are the same in value.

Two expressions are logically equivalent if they always yield the same truth values.

For instance, the statement \( p \rightarrow q \) is logically equivalent to \( \sim p \lor q \).

• This means "if \( p \), then \( q \)" is the same as "either not \( p \) or \( q \)."
• Despite their differences, these statements always resolve to the same truth value for any given inputs of \( p \) and \( q \).

Finding logical equivalencies simplifies complex statements. By using known equivalences like \( (p \rightarrow q) \leftrightarrow (\sim q \rightarrow \sim p) \), we can more easily manipulate and understand logical problems, even turning them into tautologies for easier verification.

Fallacy

A fallacy is a false statement or argument that seems valid but is fundamentally flawed. In logic, a specific kind of fallacy occurs when a statement or combination of propositions consistently leads to falsehood.

Take the statement \( (p \wedge \sim q) \wedge (\sim p \wedge q) \). It appears to make sense, but deeper analysis shows it's a fallacy.

• The problem arises because the components contradict each other: one requires \( p \) to be true and \( q \) to be false, the other requires \( \sim p \) (not \( p \)) to be true and \( q \) to be true simultaneously.
• Such conditions can never be satisfied at the same time, leading to a false outcome in every scenario.

Acknowledging fallacies is crucial in logical reasoning to separate valid arguments from those that are inherently inconsistent or misleading.