Problem 39 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The onlystatement among the following that is a tautology is \([ \mathrm{RS}]\) (a) \(A \wedge(A \vee B)\) (b) \(A \vee(A \wedge B)\) (c) \([\mathrm{A} \wedge(\mathrm{A} \rightarrow \mathrm{B})] \rightarrow \mathrm{B} \quad\) (d) \(\mathrm{B} \rightarrow[\mathrm{A} \wedge(\mathrm{A} \rightarrow \mathrm{B})]\)

Step-by-Step Solution

Answer

Option (c) is the tautology: \([A \wedge (A \rightarrow B)] \rightarrow B\).

Step 1: Analyzing Option (a)

Option (a) is \(A \wedge (A \vee B)\). According to the distributive laws, \(A \wedge (A \vee B)\) simplifies to \(A\). Since \(A\) is not always true for every possible truth value (it depends on the truth value of \(A\)), it cannot be a tautology.

Step 2: Analyzing Option (b)

Option (b) is \(A \vee (A \wedge B)\). Using the absorption law, \(A \vee (A \wedge B)\) simplifies to \(A\). Similar to option (a), since \(A\) is not always true irrespective of its truth value, this is not a tautology.

Step 3: Analyzing Option (c)

Option (c) is \([A \wedge (A \rightarrow B)] \rightarrow B\). This can be rewritten using the implication as \([A \wedge (\lnot A \vee B)] \rightarrow B\). It further simplifies to \([A \wedge B] \rightarrow B\), which is always true because if \(A \wedge B\) is true, \(B\) must be true, making the implication valid in all cases. Hence, it is a tautology.

Step 4: Analyzing Option (d)

Option (d) is \(B \rightarrow [A \wedge (A \rightarrow B)]\). Rewriting using the implication, we have \(B \rightarrow [A \wedge (\lnot A \vee B)]\) which simplifies to \(B \rightarrow (A \wedge B)\). This expression is not necessarily a tautology because when \(B\) is true, \(A\) need not necessarily be true, making \(A \wedge B\) false, which invalidates the implication in some cases.

Key Concepts

Logical Expressions

Logical expressions are mathematical expressions used in logic to represent propositions or statements. These expressions consist of variables, logical operators, and constants that collectively describe the logic of particular scenarios. Logical expressions can define a variety of logical relationships and outcomes. In this context, a tautology is a specific type of logical expression that is always true, regardless of the assigned truth values of the variables involved.

For example:

• The expression \(A \wedge (A \vee B)\) uses logical conjunction \(\wedge\) and disjunction \(\vee\) to combine propositions.
• The expression \(A \vee (A \wedge B)\), which applies the absorption law, simplifies to another logical expression.

Logical expressions allow the use of logic to solve problems and verify the truth value of complex propositions, making them crucial in computer science and mathematics.

Truth Tables

Truth tables are valuable tools used to list all possible truth values of a logical expression. They help in understanding how variables and logical operators combine to produce true or false outcomes. Each row in the table corresponds to a possible configuration of truth values for the involved variables, and the resulting truth value of the expression is computed for each configuration.

Consider the expression \([A \wedge (A \rightarrow B)] \rightarrow B\). To understand its validity as a tautology, a truth table can be constructed where:

• If \(A\) is true and \(B\) is false, \(A \rightarrow B\) is false, making \(A \wedge (A \rightarrow B)\) false, satisfying the implication \([A \wedge (A \rightarrow B)] \rightarrow B\).
• In all other cases, the expression results in true, confirming its status as a tautology.

Truth tables can therefore simplify the analysis of logical expressions, ensuring the comprehensive evaluation of all possible cases.

Logical Simplification

Logical simplification involves reducing complex logical expressions to their simplest form. This process often makes it easier to understand the core logical relationships and evaluate their truth values. Applying laws such as distributive, absorption, or De Morgan's laws can significantly streamline logical expressions.

For instance:

• In the expression \(A \wedge (A \vee B)\), using the distributive law results in \(A\).
• Further, the expression \(A \vee (A \wedge B)\) can be simplified to \(A\) using the absorption law.

By simplifying, it not only aids in assessing whether an expression is a tautology, like option (c) in the exercise, but also in eliminating redundancy and clarifying logical pathways. Simplification is a valuable skill in both academic studies and practical applications in computer science and mathematics.