Problem 10 - A Complete Resource Book in Mathematics for JEE Main

Question

Negation of "'Manu is in class X or Anu is in class XII"is (A) Manu is not in class \(\mathrm{X}\) but Anu is in class XII. (B) Manu is not is class X but Anu is not in class XII. (C) Either Manu is not in class \(\mathrm{X}\) or Anu is not is class XII. (D) None of these

Step-by-Step Solution

Answer

Option B: Manu is not in class X but Anu is not in class XII.

Step 1: Understand the Original Statement

The original statement is "Manu is in class X or Anu is in class XII." In logical terms, this is represented as \( P \lor Q \), where \( P \) is "Manu is in class X" and \( Q \) is "Anu is in class XII."

Step 2: Apply De Morgan's Laws to Negate the Statement

According to De Morgan's laws, the negation of \( P \lor Q \) is \( eg P \land eg Q \). So, the negation of the statement is "Manu is not in class X and Anu is not in class XII."

Step 3: Match the Negated Statement with the Options

The negated statement "Manu is not in class X and Anu is not in class XII" corresponds to option B: "Manu is not in class X but Anu is not in class XII."

Key Concepts

De Morgan's Laws

De Morgan's Laws are fundamental rules in logic that describe the relationships between conjunctions (AND) and disjunctions (OR) when they are negated. These laws are named after the mathematician Augustus De Morgan. The laws assist in transforming logical expressions and become especially vital when dealing with complex logical statements. Consider two logical statements: \( P \) and \( Q \).

• The first De Morgan's Law states that the negation of a conjunction \( eg (P \land Q) \) is equivalent to the disjunction of the negations \( eg P \lor eg Q \). This means if it is not true that both \( P \) and \( Q \) are true, then either \( P \) is false, or \( Q \) is false (or both).
• The second De Morgan's Law asserts that the negation of a disjunction \( eg (P \lor Q) \) results in the conjunction of the negations \( eg P \land eg Q \). Hence, if it is not the case that at least one of \( P \) or \( Q \) is true, both \( P \) must be false and \( Q \) must be false.

These transformations are key in logic and computer science, where simplifying statements is crucial.

Negation in Logic

In logic, negation is a fundamental operation that inverts the truth value of a statement. If a proposition \( P \) is true, then its negation, written as \( eg P \), is false, and vice versa. Negation changes an affirmation into a denial.

• The negation of a simple statement like "The sky is blue" becomes "The sky is not blue." In Boolean logic, if the proposition is true (\( 1 \)), its negation becomes false (\( 0 \)), and if it's false (\( 0 \)), the negation is true (\( 1 \)).
• Negation plays a significant role in constructing truth tables, which are a cornerstone of understanding complex logical expressions.

In practice, negation is often used to formulate the negative form of complex statements, making it an essential tool for logic and mathematics.

Logical Statements

Logical statements are sentences that can be either true or false. They form the basis of logical reasoning and are also used in computer science, mathematics, and philosophy to construct arguments and proofs.

• Each logical statement is usually denoted by a propositional variable, like \( P \) or \( Q \), to simplify expressions and calculations.
• Logical connectives are used to combine multiple statements. These include conjunction (AND, \( \land \)), disjunction (OR, \( \lor \)), implication (IF...THEN, \( \rightarrow \)), and equivalence (IF AND ONLY IF, \( \leftrightarrow \)).
• Understanding of logical statements allows us to evaluate real-world situations and perform logical deduction.

Logical statements can be quite complex but breaking them down into these fundamental components helps simplify and understand any logical problem.