Problem 10 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Negation of "Manu is in class \(\mathrm{X}\) or Anu is in class XII" is (A) Manu is not in class X but Anu is in class XII. (B) Manu is not is class \(\mathrm{X}\) 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

(B) Manu is not is class X but Anu is not in class XII.

Step 1: Understand the Statement

The original statement is: "Manu is in class X or Anu is in class XII." This statement can be represented using logical symbols 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 Law

To find the negation of the statement \( P \lor Q \), we use De Morgan's Laws, which state that \( eg (P \lor Q) \equiv eg P \land eg Q \). This means the negation of the original statement is: "Manu is not in class X and Anu is not in class XII." This can be written as: \( eg P \land eg Q \).

Step 3: Identify the Correct Answer

Compare the derived negated statement "Manu is not in class X and Anu is not in class XII" with the given options. This negation does not match option (A) or (C). Matches option (B) exactly: "Manu is not in class X but Anu is not in class XII."

Step 4: Conclusion

The correct negation has been found in option (B). Thus, the answer to the question is (B).

Key Concepts

De Morgan's Laws

Understanding De Morgan's Laws is crucial for grasping logical reasoning in mathematics. These laws are fundamental rules in set theory and mathematical logic, named after the 19th-century logician Augustus De Morgan. They describe how the negation of a conjunction or disjunction of two statements can be expressed as the negation of each statement, using opposite operators. Here's a quick look:

• If you have a disjunction (an "or" statement) like \( A \lor B \), its negation will be equivalent to the conjunction of the negations: \( eg A \land eg B \).
• Conversely, if you begin with a conjunction (an "and" statement) like \( A \land B \), the negation will become a disjunction of the negations: \( eg A \lor eg B \).

In our exercise, the statement "Manu is in class X or Anu is in class XII" can be denoted as \( P \lor Q \). By applying De Morgan's Laws, its negation becomes \( eg P \land eg Q \), meaning "Manu is not in class X and Anu is not in class XII."

De Morgan's Laws help us systematically negate complex statements, making them essential tools in logic.

Logical Negation

Logical negation is a basic operation in mathematical logic that inverts the truth value of a statement. When we negate a proposition, if the original statement is true, the negation becomes false, and vice-versa. In symbols, negation is often represented as \( eg \).

Let's take statement \( P \), "Manu is in class X":

• The negation is \( eg P \), which reads "Manu is not in class X."

Similarly, for statement \( Q \), "Anu is in class XII":

• The negation is \( eg Q \), "Anu is not in class XII."

When dealing with compound statements like "\( P \lor Q \)" ("Manu is in class X or Anu is in class XII"), negation requires the use of De Morgan's Laws.

Logical negation is a critical tool for developing a deeper understanding of mathematical arguments and for constructing logically sound conclusions.

Mathematical Logic

Mathematical logic forms the foundation of much of mathematical reasoning and is essential for understanding topics like computation, set theory, and algorithms. It deals with formal systems that define logical relations and provides tools to analyze and construct arguments.

Key components in mathematical logic include:

• Propositions: Statements that are either true or false.
• Operators: Symbols such as "and" (\( \land \)), "or" (\( \lor \)), and "not" (\( eg \)) that are used to form compound statements.
• Proofs: Logical arguments demonstrating a proposition's truth.
• Inference Rules: Guidelines for deriving conclusions from premises.

In our exercise, mathematical logic is applied to transform and analyze statements, like using De Morgan's Laws to find the negation. Understanding logic helps in various fields, from computer science to mathematics, making it a versatile and vital discipline.