Problem 52 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Contrapositive of the statement "If two numbers are not equal, then their squares are not equal", is : (a) If the squares of two numbers are not equal, then the numbers are equal. (b) If the squares of two numbers are equal, then the numbers are not equal. (c) If the squares of two numbers are equal, then the numbers are equal. (d) If the squares of two numbers are not equal, then the numbers are not equal.

Step-by-Step Solution

Answer

(c) If the squares of two numbers are equal, then the numbers are equal.

Step 1: Understanding the Statement

The given statement is "If two numbers are not equal, then their squares are not equal." This can be represented logically as: If \( a eq b \) then \( a^2 eq b^2 \).

Step 2: Definition of Contrapositive

The contrapositive of a statement "If \( p \), then \( q \)" is "If not \( q \), then not \( p \)." This always holds the same truth value as the original statement.

Step 3: Applying Contrapositive

For the given statement "If \( a eq b \), then \( a^2 eq b^2 \)," the contrapositive will be "If \( a^2 = b^2 \), then \( a = b \)."

Step 4: Identifying the Correct Answer

Comparing our contrapositive "If \( a^2 = b^2 \), then \( a = b \)" with the options given, we find that option (c) "If the squares of two numbers are equal, then the numbers are equal" matches our contrapositive.

Key Concepts

Logical Reasoning

Logical reasoning forms the core of mathematical problem-solving. It involves the use of structured logic to arrive at a valid conclusion. For instance, the idea of "if-then" statements, also known as conditional statements, often pop up in logical reasoning.
These statements set a condition for something to happen, laying down a scenario and its possible outcomes.
When engaging in logical reasoning, you:

• Identify the premises, which are the facts or assumptions you start with.
• Understand the logical relationships between these premises.
• Determine the conclusions that can be logically drawn.

Understanding contrapositive statements is crucial as they can often simplify complex logical connections.
They help in "flipping" a statement to examine its truth from another angle, allowing for more comprehensive reasoning.

Mathematical Logic

Mathematical logic is the framework that underpins logical reasoning in mathematics.
It involves a formal, mathematical study of logic, which is the basis for proofs and theorems.
This study encompasses structures like sets, sequences, and functions, which help in formalizing mathematical statements.
Use of symbols, such as \( \rightarrow \) for conditional statements, \( \wedge \) for 'and', and \( eg \) for negation, helps in expressing and analyzing mathematical logic clearly.
Mathematicians rely on this to understand relations and firm conclusions about mathematical operations.
Mathematical logic allows us to:

• Determine truth values of statements.
• Formulate theorems.
• Create proofs.

High-level mathematical reasoning involves proving, disapproving, or exploring the truth value of these statements in various contexts.

Conditional Statements

Conditional statements are a vital component of both logical reasoning and mathematical logic.
They typically follow the format "If \( p \), then \( q \)," meaning the truth of \( q \) depends on the truth of \( p \).
These statements are further analyzed through their inverses, converses, and contrapositives.
A contrapositive statement takes the original "if-then" format and flips and negates it, becoming "If not \( q \), then not \( p \)."

• The contrapositive always holds the same truth value as the original conditional statement.
• This property makes contrapositives extremely useful in proofs and problem-solving.
• Identifying the contrapositive is a common technique to validate or prove statements in mathematics.

By learning to correctly formulate and interpret conditional statements, students can develop sharper logical thinking and better problem-solving skills.