Problem 9 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The logical statement \((\mathrm{p} \Rightarrow \mathrm{q}) \wedge(\mathrm{q} \Rightarrow \sim \mathrm{p})\) is equivalent to: \(\quad\) (a) \(p\) (b) \(q\) (c) \(\sim p\) (d) \(\sim q\)

Step-by-Step Solution

Answer

The logical statement is equivalent to \(\sim p\), choice (c).

Step 1: Understanding Implication

The logical implication \(p \Rightarrow q\) can be rewritten using disjunction: \(\sim p \lor q\). Similarly, \(q \Rightarrow \sim p\) can be rewritten as \(\sim q \lor \sim p\). Recognizing these transformations will help simplify the expression.

Step 2: Rewriting the Original Expression

Rewrite the original statement \((p \Rightarrow q) \wedge (q \Rightarrow \sim p)\) as \((\sim p \lor q) \wedge (\sim q \lor \sim p)\). This form will allow us to apply logical simplification rules more easily.

Step 3: Applying Distributive Law

Apply the distributive law to \((\sim p \lor q) \wedge (\sim q \lor \sim p)\). Distribute to obtain: \( ((\sim p \lor q) \wedge \sim q) \lor ((\sim p \lor q) \wedge \sim p)\).

Step 4: Simplifying the Expression

Evaluate each part of the distributed expression separately:- For \((\sim p \lor q) \wedge \sim q\), use the fact that \(q\) and \(\sim q\) can't both be true. Thus, this simplifies to \(\sim p\).- For \((\sim p \lor q) \wedge \sim p\), the presence of \(\sim p\) in both terms means this also simplifies to \(\sim p\).Thus, the entire expression simplifies to \(\sim p\).

Step 5: Determining the Equivalent Statement

Based on the simplification, the expression \((p \Rightarrow q) \wedge (q \Rightarrow \sim p)\) is equivalent to \(\sim p\). Therefore, the correct choice is option (c).

Key Concepts

Implication in Logic

In logic, implications are a fundamental concept often noted by the symbol "\(\Rightarrow\)". The expression \(p \Rightarrow q\) (read as "\(p\) implies \(q\)") means that if \(p\) is true, then \(q\) must also be true.
Here’s where it gets interesting for logical equivalence: an implication can actually be rewritten in terms of disjunction ("or" statements), which is helpful for simplification purposes. This transformation is captured by the equivalence \(p \Rightarrow q\) being equal to \(\sim p \lor q\).
This insight allows us to cleverly manipulate logical expressions by switching between implications and disjunctions, thus aiding in simplification and further analysis of logical statements.

Logical Simplification

Simplifying logical expressions involves reducing complex statements to more manageable forms without altering their truth values. This is similar to factorizing expressions in algebra.
To simplify, we often use transformations, like rewriting implications as disjunctions. Our original expression

• \((p \Rightarrow q) \wedge (q \Rightarrow \sim p)\)

becomes:

• \((\sim p \lor q) \wedge (\sim q \lor \sim p)\).

Using such transformations, we prepare the expression for further deduction using other logical equivalences and laws, such as the distributive law. This can dramatically simplify our work and make complex logical problems more manageable.

Distributive Law in Logic

The distributive law is a familiar concept in both algebra and logic. In logic, it allows us to distribute conjunctions and disjunctions over each other, much like we distribute multiplication over addition in arithmetic.
For example, applying the distributive law to:

• \((\sim p \lor q) \wedge (\sim q \lor \sim p)\)

we arrive at:

• \(((\sim p \lor q) \wedge \sim q) \lor ((\sim p \lor q) \wedge \sim p)\).

This method can unravel complex expressions, allowing for step-by-step simplification and evaluation of each component part.
With this approach, contradictions like \(q \wedge \sim q\) are easily identified and simplified, which paves the way for determining equivalent logical expressions in a straightforward manner.