Problem 57 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Consider the following statements \(\mathrm{P}\) : Suman is brilliant \(\mathrm{Q}\) : Suman is rich. \(\mathrm{R}\) : Suman is honest the negation of the statement "Suman is brilliant and dishonest if and only if suman is rich" can be equivalently expressed as (a) \(\sim Q \leftrightarrow \sim \vee \mathbb{R}\) (b) \(\sim Q \leftrightarrow \sim \vec{P} \wedge R\) \((c) \sim Q \leftrightarrow P \vee \sim R\) \((\mathrm{d}) \sim \mathrm{Q} \leftrightarrow \mathrm{P} \wedge \sim \mathrm{R}\)

Step-by-Step Solution

Answer

(d) \(\sim Q \leftrightarrow P \wedge \sim R\).

Step 1: Understanding the Initial Statement

The initial statement is given: 'Suman is brilliant and dishonest if and only if Suman is rich'. This can be transformed into a logical expression using the statements: \( P \): Suman is brilliant, \( Q \): Suman is rich, and \( R \): Suman is honest. Therefore, the statement can be written as \(( P \wedge \sim R ) \leftrightarrow Q\).

Step 2: Determine the Negation

We need to negate the initial logical statement. The negation of \(( A \leftrightarrow B )\) is \(( A \leftrightarrow B ) \equiv \sim ( A \leftrightarrow B )\). Thus, the negation of \(( P \wedge \sim R ) \leftrightarrow Q\) is \( \sim ( ( P \wedge \sim R ) \leftrightarrow Q )\).

Step 3: Apply the Negation of Biconditional

The negation of a biconditional \(( A \leftrightarrow B )\) results in \(( A \wedge \sim B ) \vee ( \sim A \wedge B )\). This changes the problem to \(( P \wedge \sim R \wedge \sim Q ) \vee ( \sim ( P \wedge \sim R ) \wedge Q )\).

Step 4: Simplify Each Part

Simplifying \(\sim ( P \wedge \sim R )\) results in \(\sim P \vee R\) using De Morgan's Law. Therefore, the expression becomes \(( P \wedge \sim R \wedge \sim Q ) \vee ( (\sim P \vee R) \wedge Q )\).

Step 5: Check Equivalent Expression

Now, check which of the given options match the derived expression for the negation. The correct simplified equivalent form from the options is (d): \(\sim Q \leftrightarrow P \wedge \sim R\). This matches the transformed and simplified expression.

Key Concepts

Logical Equivalence

Logical equivalence is a fundamental concept in logic that implies two statements or expressions lead to the same truth value in any scenario. When we say that two statements are logically equivalent, we're asserting that no matter what the context or scenario, both statements will either be true or false at the same time.
Logical equivalence is often represented using the symbol \( \equiv \). For example, the expressions \( A \wedge B \equiv B \wedge A \) demonstrate logical equivalences due to the commutative property of logical conjunction.
Recognizing logical equivalence is crucial because it allows us to transform complex statements into more manageable forms without altering the fundamental truth they represent. This is particularly useful in simplifying expressions and solving logical problems, such as finding the negation of a statement while preserving its meaning.

Negation of Statements

Negation plays a vital role in logic as it reverses the truth value of a given statement. If a statement is true, its negation is false, and vice versa. In logical expressions, negation is typically represented by the symbol \( \sim \).
For example, if \( P \) denotes 'It is raining,' then \( \sim P \) signifies 'It is not raining.' Understanding how negation works is crucial, especially when dealing with conditional and biconditional statements.
Let's consider the procedure for negating a biconditional statement \( A \leftrightarrow B \). The negation, \( \sim (A \leftrightarrow B) \), can be expanded into two parts: the situation where \( A \) is true and \( B \) is false, and the reverse case, where \( A \) is false, and \( B \) is true. Thus, it breaks down to \((A \wedge \sim B) \vee (\sim A \wedge B)\). This allows us to express the negated version of a biconditional in a form that reveals more about the true conditions and exceptions.

Boolean Algebra

Boolean algebra is a branch of algebra that deals with boolean values, typically represented as true and false (or 1 and 0). In logical terms, it allows us to perform operations that apply logical reasoning over these values.
The basic operations in Boolean algebra include conjunction (AND, \( \wedge \)), disjunction (OR, \( \vee \)), and negation (NOT, \( \sim \)). These operations follow specific laws and properties such as De Morgan's Laws, which help in simplifying logical expressions.
De Morgan's Laws are particularly helpful when working with negations, as they provide a way to distribute a negation across conjunctions and disjunctions. For instance, De Morgan's Laws state that \( \sim (A \wedge B) \equiv \sim A \vee \sim B \) and \( \sim (A \vee B) \equiv \sim A \wedge \sim B \). Mastery of these concepts allows students to transform and simplify complex logical statements, making problem-solving more efficient and intuitive.