Problem 50 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

For any two statements \(\mathrm{p}\) and \(\mathrm{q}\), the negation of the expression \(\mathrm{p} \vee(\sim \mathrm{p} \wedge \mathrm{q})\) is: (a) \(\sim \mathrm{p} \wedge \sim \mathrm{q}\) (b) \(\mathrm{p} \wedge q\) (c) \(\mathrm{p} \leftrightarrow q\) (d) \(\sim \mathrm{p} \vee \sim \mathrm{q}\)

Step-by-Step Solution

Answer

The answer is (a) \( \sim \mathrm{p} \wedge \sim \mathrm{q} \).

Step 1: Understand the Expression

The expression to negate is \( \mathrm{p} \vee (\sim \mathrm{p} \wedge \mathrm{q}) \). It consists of the disjunction (OR) of \( \mathrm{p} \) with \( \sim \mathrm{p} \wedge \mathrm{q} \).

Step 2: Apply De Morgan's Laws

Applying De Morgan's laws to the expression \( \sim(\mathrm{p} \vee (\sim \mathrm{p} \wedge \mathrm{q})) \) gives the equivalent expression \( \sim \mathrm{p} \wedge \sim (\sim \mathrm{p} \wedge \mathrm{q}) \). This step involves negating the entire disjunction.

Step 3: Simplify Inside the Conjunction

Next, we simplify the negation inside the conjunction: \( \sim (\sim \mathrm{p} \wedge \mathrm{q}) \) becomes \( \mathrm{p} \vee \sim \mathrm{q} \) using De Morgan's laws again.

Step 4: Combine Parts

Combine the results from steps 2 and 3 into \( \sim \mathrm{p} \wedge (\mathrm{p} \vee \sim \mathrm{q}) \). Use distribution to expand this to \((\sim \mathrm{p} \wedge \mathrm{p}) \vee (\sim \mathrm{p} \wedge \sim \mathrm{q})\).

Step 5: Simplify Using Contradictions

Since \( \sim \mathrm{p} \wedge \mathrm{p} \) is a contradiction, it simplifies to \( \text{False} \). Therefore, the expression reduces to \( \sim \mathrm{p} \wedge \sim \mathrm{q} \).

Step 6: Identify the Correct Answer

The simplified expression \( \sim \mathrm{p} \wedge \sim \mathrm{q} \) matches option (a).

Key Concepts

De Morgan's laws

De Morgan's laws are fundamental rules in logic that are used to simplify and transform logical expressions. They connect two types of basic logical operations: negation and conjunction/disjunction. Understanding these laws helps you easily manipulate expressions involving AND (\(\wedge\)) and OR (\(\vee\)) operators, especially when negation is involved. Here’s a quick breakdown of the laws:

• De Morgan's first law states that the negation of a conjunction is equivalent to the disjunction of the negations. Formally, this can be written as \( \sim(p \wedge q) = \sim p \vee \sim q \).
• De Morgan's second law states that the negation of a disjunction is equivalent to the conjunction of the negations: \( \sim(p \vee q) = \sim p \wedge \sim q \).

These laws are applied in logical expressions to simplify them by changing the logical operations and their negations in a structured way. When faced with an expression, these laws are instrumental in reshaping it, thus helping to find logical equivalences or simplifying complex logical statements.

Logical Expressions

A logical expression is composed of variables, constants, and logical operations that convey logical statements and propositions. These expressions are integral to fields like mathematics, computer science, and philosophy, as they form the basis of Boolean logic which underpins modern computing. Logical expressions commonly use logical connectives such as:

• AND (\(\wedge\))
• OR (\(\vee\))
• NOT (\(\sim\))

These operations allow the combination and modification of basic logical assertions into more complex expressions, which can be evaluated for truth values depending on the propositions involved. For instance, in the expression \(\mathrm{p} \vee (\sim \mathrm{p} \wedge \mathrm{q})\), we have a mix of propositions (\(\mathrm{p}\) and \(\mathrm{q}\)) and logical operations (OR and AND) nested with negation. Simplifying such expressions often involves using previously-mentioned laws, like De Morgan's laws, to transform and simplify for clarity or specific outcomes.

Disjunction and Conjunction

Disjunction and conjunction are two primary binary operations in logical expressions, often seen in expressions and conditions in mathematical logic.**Disjunction:** The disjunction (OR operation), symbolized by \(\vee\), combines two propositions to form a new proposition that is true if at least one of the propositions is true. For example, in \(\mathrm{p} \vee \mathrm{q}\), the expression is true if either \(\mathrm{p}\) or \(\mathrm{q}\) is true or both.**Conjunction:** The conjunction (AND operation), symbolized by \(\wedge\), combines two propositions such that the resulting proposition is only true if both propositions are true. For example, \(\mathrm{p} \wedge \mathrm{q}\) is only true if both \(\mathrm{p}\) and \(\mathrm{q}\) are true.When working with these operations:

• Remember that disjunction allows more flexibility in meeting the truth than conjunction.
• These operations are often coupled with negation, and simplification might require the use of De Morgan's laws to transform expressions, especially when expressions get nested or more complex.

Understanding and mastering these concepts are crucial in constructing, interpreting, and manipulating logical expressions to derive accurate logical conclusions.