Problem 24 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

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

Step-by-Step Solution

Answer

None of the given expressions is a tautology.

Step 1: Define the term "tautology"

A tautology is a logical statement that is always true, no matter what the truth values of the individual components are. To determine if a given logical expression is a tautology, we substitute all possible truth values for the variables and check if the expression remains true in all cases.

Step 2: Construct truth tables for each option

To check which expression is a tautology, we need to create truth tables for all options. The variables commonly involved are \( p \), \( q \), and \( \sim \) (negation), as well as \( \wedge \) (AND), \( \vee \) (OR), and \( \rightarrow \) (implies). We evaluate each column of the expression based on all possible truth values (true or false) of \( p \) and \( q \).

Step 3: Evaluate option (a)

For (a), evaluate \((\sim p) \wedge (p \vee q) \rightarrow q\). This involves computing \(\sim p\), then \(p \vee q\), then applying \(\wedge\), and finally the implication \(\rightarrow q\). After constructing the full truth table, it's found that not all possible input values yield true results.

Step 4: Evaluate option (b)

For (b), the expression is \((q \rightarrow p) \vee \sim (p \rightarrow q)\). First, compute \(q \rightarrow p\) and \(p \rightarrow q\), then find \(\sim (p \rightarrow q)\), and subsequently evaluate the OR operation \(\vee\). This table also results in elements that are not consistently true for all cases.

Step 5: Evaluate option (c)

For (c), \((\sim q) \vee (p \wedge q) \rightarrow q\). Calculate \(\sim q\), \(p \wedge q\), evaluate the OR \(\vee\), and then the final implication \(\rightarrow q\). Again, this expression fails to be true in all scenarios.

Step 6: Evaluate option (d)

For (d), evaluate \((p \rightarrow q) \wedge (q \rightarrow p)\). Calculate each implication individually (which will be true when both \(p\) and \(q\) have the same truth values and false otherwise) and then apply the AND operation \(\wedge\). This whole expression, however, only provides truth when both conditions are met.

Step 7: Conclude the tautology

Upon building truth tables for all expressions (a, b, c, d), evaluate each row to see if they consistently produce a true result for every combination of \( p \) and \( q \). In this instance, none of the options is a tautology as none gives exclusively true results in all situations.

Key Concepts

Truth Tables

Truth tables are crucial tools in logic that help us evaluate the truthfulness of logical statements based on all possible combinations of truth values. They consist of columns that list each variable and logical operator used in a statement. For example, in a statement involving two variables, such as \( p \) and \( q \), we would have four rows representing each combination of truth values for these variables:

• (True, True)
• (True, False)
• (False, True)
• (False, False)

In a truth table, each row represents one possible scenario. We fill out the table by determining the result of the logical expression's subcomponents, ultimately arriving at whether the entire expression is true or false given those initial conditions. This process is particularly helpful when identifying tautologies, as they require the final column to yield true for every row. Truth tables visually demonstrate the function of logical connectives and help in proving whether an expression is valid universally.

Logical Statements

Logical statements are expressions that can be true or false. These statements are fundamental in logic and are often composed using variables and logical connectives like AND, OR, and NOT.
They allow us to construct more complex expressions to describe conditions and relations in a precise manner. In the context of this exercise, statements are examined to determine whether they are tautologies - statements that are always true, regardless of the truth values of their components.
To analyze these statements, we break them down into logical components and operators. For instance, in the expression \((p \rightarrow q) \wedge (q \rightarrow p)\), the statement relies on two logical implications combined with an AND operator. By understanding the truth values of individual components, we can calculate the overall truth of the statement.

Implication in Logic

Implication is one of the fundamental logical connectives, often represented by the symbol \(\rightarrow\). It is read as "implies" or "if...then."
In logical terms, an implication \(p \rightarrow q\) is considered false only when the premise \(p\) is true, and the conclusion \(q\) is false.
In all other cases, the statement is true. This means that:

• If \(p\) is true and \(q\) is true, \(p \rightarrow q\) is true.
• If \(p\) is false and \(q\) is true, \(p \rightarrow q\) is true.
• If \(p\) is false and \(q\) is false, \(p \rightarrow q\) is true.
• If \(p\) is true and \(q\) is false, \(p \rightarrow q\) is false.

Implication is key in evaluating the expressions given in this exercise. For instance, understanding how the implication behaves helps in determining the truthfulness of complex statements and whether they indeed qualify as tautologies.

Logical Connectives

Logical connectives are operators used to combine simple statements into more complex ones. The main logical connectives include:

• NOT (\(\sim\)) - Negates the truth value of a statement
• AND (\(\wedge\)) - True only if both operands are true
• OR (\(\vee\)) - True if at least one operand is true
• Implies (\(\rightarrow\)) - True unless a true statement implies a false statement

Each connective transforms the truth values of the statements involved and determines the result of the logical expression. Their interplay is crucial when building truth tables and evaluating whether statements are tautologies. For instance, an AND operation \((\wedge)\) will result in true only when both contributing statements are true, affecting the overall truthfulness of a more complex logical expression.