Problem 36 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Statement-1: The statement \(A \rightarrow(B \rightarrow A)\) is equivalent to \(A \rightarrow(A \vee B)\). Statement- \(2:\) The statement \(\sim[(\mathrm{A} \wedge \mathrm{B}) \rightarrow(\sim \mathrm{A} \vee \mathrm{B})]\) is a Tautology (a) Statement- 1 is false; Statement- 2 is true. (b) Statement- 1 is true; Statement- 2 is true; Statement- 2 is not correct explanation for Statement-1. (c) Statement- 1 is true; Statement- 2 is false. (d) Statement- 1 is true; Statement- 2 is true; Statement- 2 is the correct explanation for Statement-1.

Step-by-Step Solution

Answer

Statement 1 is true, Statement 2 is false. Answer is (c).

Step 1: Analyze Statement 1

We need to check the equivalence between \(A \rightarrow (B \rightarrow A)\) and \(A \rightarrow (A \vee B)\). - \(A \rightarrow (B \rightarrow A)\) simplifies to \(\sim A \vee (\sim B \vee A)\) which further simplifies to \(\sim A \vee A \vee \sim B\), and this simplifies to \(\top\) (tautology) based on the principle that \(\sim A \vee A\) is a tautology.- \(A \rightarrow (A \vee B)\) simplifies to \(\sim A \vee (A \vee B)\), which also simplifies to \(\top\) because \(\sim A \vee A\) is a tautology.Both statements are tautologies, hence they are equivalent.

Step 2: Evaluate Statement 2

We need to check if \(\sim[(A \wedge B) \rightarrow (\sim A \vee B)]\) is a tautology. - \((A \wedge B) \rightarrow (\sim A \vee B)\) simplifies to \(\sim(A \wedge B) \vee (\sim A \vee B)\), which simplifies to \((\sim A \vee \sim B) \vee \sim A \vee B\) which simplifies to \((\sim A \vee \sim A) \vee (\sim B \vee B)\), thus reducing to \(\sim A \vee \top\), which is \(\top\).- Therefore, the original expression becomes \(\sim \top\), which is \(\bot\) (not a tautology). Statement 2 is false.

Key Concepts

Logical Equivalence

In propositional logic, logical equivalence refers to when two statements are always the same in truth value, meaning they yield identical results in all possible scenarios. To determine equivalence, we rely on simplifying logical expressions and checking if they produce the same outcomes.

For example, the exercise shows that the statement \( A \rightarrow (B \rightarrow A) \) can be broken down and examined as \( \sim A \vee (\sim B \vee A) \). Simplifying further, this becomes \( \sim A \vee A \vee \sim B \), which is recognized as always true due to the laws of logic, specifically the fact that \( \sim A \vee A \) is a tautology. Similar steps are taken with \( A \rightarrow (A \vee B) \) revealing it also simplifies to a tautology.

Thus, these statements are equivalent since both simplify to a tautology, showing their truth values align in every possible scenario. Understanding logical equivalence helps in determining when different expressions effectively communicate the same logical relationship.

Tautology

A tautology in logic is a formula or assertion that is always true, no matter what the truth values of the individual components are. Recognizing tautologies is crucial because they reflect logical truths that are universally valid.

Taking an example from the exercise: \( A \rightarrow (A \vee B) \) simplifies to \( \sim A \vee (A \vee B) \). Here, \( \sim A \vee A \) signifies a tautology since it is always true, regardless of \( A \)'s truth value. The entire expression simplifies to a tautology, confirming the statement’s justified truth under any condition.

In contrast, evaluating \( \sim[(A \wedge B) \rightarrow (\sim A \vee B)] \) involves transforming it to \( \sim \top \), resulting in \( \bot \) (false), which is not a tautology. This part of the exercise highlights the evaluation method to determine logical validity.

Logical Connectives

Logical connectives are symbols used in logic to connect statements or propositions. They help in constructing complex expressions by combining simpler ones. Common logical connectives include \( \wedge \) (and), \( \vee \) (or), \( \sim \) (not), \( \rightarrow \) (implies), and \( \equiv \) (if and only if).

In the given exercise, the connectives \( \rightarrow \), \( \vee \), and \( \sim \) play crucial roles. The expression \( A \rightarrow B \) is translated into \( \sim A \vee B \), demonstrating how implication can be rephrased using 'or' and 'not'. This transformation is crucial for simplifying expressions and checking equivalence or tautology.

Logical connectives allow us to build complex, logical relationships from simple statements, and understanding them is key to resolving logical problems efficiently. Mastery of these connectives enables one to parse and understand logical expressions more clearly.