Problem 8 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Which of the following statements is a tautology? (a) \(p \vee(\sim q) \rightarrow p \wedge q\) (b) \(\sim(p \wedge \sim q) \rightarrow p \vee q\) (c) \(\sim(p \vee \sim q) \rightarrow p \wedge q\) (d) \(\sim(p \vee \sim q) \rightarrow p \vee q\)

Step-by-Step Solution

Answer

Option (d) is a tautology.

Step 1: Understanding Tautology

A statement is a tautology if it is true in every possible interpretation. Our task is to evaluate each option to determine if it's a tautology.

Step 2: Analyze Option (a)

Proposal for (a): Analyze the statement \( p \vee (\sim q) \rightarrow p \wedge q \). To check if it is a tautology, we need to create a truth table and determine if it is true in all cases. For cases where \( p \) is false, the statement is false. Therefore, it is not a tautology.

Step 3: Analyze Option (b)

Proposal for (b): Analyze \( \sim(p \wedge \sim q) \rightarrow p \vee q \). Construct a truth table and check each possible scenario. For any input Combination \((p = \text{False}, q = \text{True})\) the statement is true, and it holds true for all input combinations. Therefore, it seems to be always true.

Step 4: Analyze Option (c)

Proposal for (c): Analyze \( \sim(p \vee \sim q) \rightarrow p \wedge q \). Construct a truth table. For \( p = \text{False} \) and \( q = \text{True} \), the statement does not hold, making it not a tautology.

Step 5: Analyze Option (d)

Proposal for (d): Analyze \( \sim(p \vee \sim q) \rightarrow p \vee q \). With a truth table, you find that there is no scenario where the statement evaluates to false, meaning it is true for all combination of \(p\) and \(q\). Therefore, it is a tautology.

Step 6: Conclusion

Among all options analyzed, the statement in option (d) is true in every possible scenario, verifying that it is indeed a tautology.

Key Concepts

Truth Table

A truth table is a simple way to analyze logical expressions by listing all possible combinations of truth values for each variable. It helps determine whether a statement is always true, sometimes true, or never true. The table consists of columns for each variable and additional columns for each part of the logical statement.

• The first step is to identify all variables involved in the expression.
• The second step is to systematically list all possible combinations of truth values (true or false).
• Finally, evaluate the logic of the expression for each row in the table to determine the outcome.

Creating a truth table involves an organized approach that allows you to see clearly whether a statement holds true under all possible scenarios. This method is essential for evaluating complex logical statements and particularly useful when examining potential tautologies.

Logical Statements

Logical statements are expressions made up of variables and logical operations like 'and', 'or', and 'not'. They form the building blocks of arguments in formal logic, often used to determine the truth of a given argument.
In mathematics and computer science, these statements often involve symbolic representations:

• (p AND q), represented as p ∧ q, is true only if both p and q are true.
• (p OR q), represented as p ∨ q, is true if at least one of p or q is true.
• (NOT p), represented as ¬p, negates the truth value of p.

Understanding these fundamental operations can help in parsing complex logical statements and in constructing truth tables for deeper insights. This insight is crucial for finding tautologies and discrepancies in logical reasoning.

Interpretation in Logic

In the context of logic, interpretation refers to the assignment of truth values to variables in a logical expression to evaluate its validity. This process is essential for determining whether a statement is a tautology.
Consider an expression like \(\sim(p \vee \sim q) \rightarrow p \vee q\).Here, you would interpret by assigning true or false values to\( p \) and \( q \) and assessing the overall truth of the statement using a truth table.
The interpretation helps to explore every possible scenario and conclude the universal truth of a statement. Thus, when a logical statement is true for all interpretations, it is considered a tautology, meaning it's infallible within the constraints of logic itself.
Understanding the nuances of interpretation in logic enables one to discern between tautologies and contingent statements, sharpening reasoning skills in formal logic.