Problem 42 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(p\) be the statement \(^{4} x\) is an irrational number", \(q\) be the statement "y is a transcendental number", and \(r\) be the statement " \(x\) is a rational number iff \(y\) is a transcendental number". \(\quad\) [2008] Statement-1: \(r\) is equivalent to either \(q\) or \(p\) Statement- \(2: r\) is equivalent to \(\sim(p \leftrightarrow \sim q)\). (a) Statement \(-1\) is false, Statement- 2 is true (b) Statement \(-1\) is true, Statement- 2 is true; Statement \(-2\) is a correct explanation for Statement- 1 (c) Statement \(-1\) is true, Statement- 2 is true; Statement \(-2\) is not a correct explanation for Statement-1 (d) Statement \(-1\) is true, Statement- 2 is false

Step-by-Step Solution

Answer

Option (a): Statement 1 is false, Statement 2 is true.

Step 1: Understand the Logical Statements

We start with understanding each statement. Statement 1 says "\(r\) is equivalent to either \(q\) or \(p\)" and Statement 2 claims "\(r\) is equivalent to \(\sim(p \leftrightarrow \sim q)\)". Here \(\sim\) denotes negation, \(\leftrightarrow\) denotes biconditional (if and only if), and \(\text{or}\) denotes alternative disjunction.

Step 2: Analyze Statement 1

For statement 1, "\(r\) is equivalent to either \(q\) or \(p\)", we can write it as \(r \equiv q \vee r \equiv p\). The biconditional \(r: x \text{ is rational } \leftrightarrow y \text{ is transcendental}\) is simplified as \((x \text{ is rational } \wedge y \text{ is transcendental}) \vee (x \text{ is irrational } \wedge y \text{ is algebraic})\). We examine if \(r\) matches either \(p\) or \(q\).

Step 3: Analyze Statement 2

Now look at Statement 2 in logical form: \(r \equiv \sim(p \leftrightarrow \sim q)\). Simplify \((p \leftrightarrow \sim q)\), which is equivalent to \((p \wedge q) \vee (\sim p \wedge \sim q)\), and \(r \equiv (\sim p \vee q) \wedge (p \vee \sim q)\). This is logically equivalent to \(r\).

Step 4: Compare Statements

After analyzing both, check if Statement 1 and Statement 2 are formulas that evaluate to true or false separately. From the analysis, the expression obtained in Statement 2 indeed matches the expression derived from \(r\), proving Statement 2 to be true. However, from Statement 1, "r is equivalent to q or p" does not hold as true, because \(r\) matches \(\sim(p \leftrightarrow \sim q)\), not separately \(p\) or \(q\).

Step 5: Conclusion Based on Analysis

To conclude, Statement 1 is false because \(r\) does not simplify solely to either \(p\) or \(q\). Statement 2 is true; \(r\) is equivalent to \(\sim(p \leftrightarrow \sim q)\). Statement 2 evaluates correctly but does not help in proving statement 1. Therefore, option (a) Statement 1 is false, Statement 2 is true.

Key Concepts

Biconditional Statements

Biconditional statements are a crucial part of logical reasoning. They are represented by the symbol \( \leftrightarrow \), which reads as "if and only if". This type of statement is true only when both constituent statements are either true or false. Essentially, both must have the same truth value.
Imagine you have two statements, say \( p \) and \( q \). The biconditional statement \( p \leftrightarrow q \) asserts that \( p \) is true if \( q \) is true, and \( p \) is false if \( q \) is false.

Some key points about biconditional statements include:

• The truth of the entire biconditional hinges on both statements agreeing in truth value.
• Biconditional is symmetric: \( p \leftrightarrow q \) is the same as \( q \leftrightarrow p \).
• Commonly used to show a direct correlation between两个 statements.

In our original problem, the statement \( r: x \text{ is rational } \leftrightarrow y \text{ is transcendental} \) can be expanded by further examining the truth chain of rational and transcendental numbers, deeply embedding the idea of mutual dependency.

Negation in Logic

Negation is a fundamental concept in logical reasoning. It is denoted by the symbol \( \sim \), and implies reversing the truth value of a statement. If \( p \) is true, then \( \sim p \) is false, and vice versa.

To understand negation:

• Negating a true statement makes it false, and vice versa.
• It is commonly used to express the inverse of a given condition or expression.
• Negation can be applied to complex logical constructs, fundamentally altering their meaning.

In the exercise, negation plays a key role in the expression \( \sim(p \leftrightarrow \sim q) \). This means we negate the biconditional statement \( p \leftrightarrow \sim q \), which fundamentally changes how we interpret the relationship between \( p \) and \( q \).
This alteration reflects how powerful negation can be in transforming a logical expression and deriving new insights.

Logical Equivalence

Logical equivalence is a vital part of understanding logical relationships. Two statements are said to be logically equivalent if they always have the same truth value, no matter the circumstances.

There are a few crucial aspects to grasp regarding logical equivalence:

• Equivalence allows us to transform complex statements into simpler or different forms, without changing their meaning.
• It often helps in proofs, particularly when we need to show that two statements lead to the same conclusion.
• Common techniques to prove equivalence include truth tables and various logical laws and rules.

In our problem, we determined that \( r \equiv \sim(p \leftrightarrow \sim q) \) is indeed true and logically equivalent, unlike the proposition that \( r \equiv q \lor p \). This highlights how logical equivalence can rectify misunderstandings by showing how statements relate or differ from one another.
Logical equivalence was crucial in solving the exercise and pointing out the correct assessment of statements.