Problem 54 - 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 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 not equal, then the numbers are not equal. (d) If the squares of two numbers are not equal, then the numbers are equal.

Step-by-Step Solution

Answer

Option (a) is the contrapositive: 'If the squares of two numbers are equal, then the numbers are equal.'

Step 1: Understand the Original Statement

The original statement we have is 'If two numbers are not equal, then their squares are not equal'. This is a conditional statement where the first part (antecedent) is 'two numbers are not equal' and the second part (consequent) is 'their squares are not equal'.

Step 2: Recall the Definition of Contrapositive

The contrapositive of a statement 'If P, then Q' is 'If not Q, then not P'. This is distinct from the converse or inverse and is logically equivalent to the original statement.

Step 3: Identify Not Q and Not P

For the statement 'If two numbers are not equal, then their squares are not equal': - Q: 'their squares are not equal' - Not Q: 'the squares of two numbers are equal' - P: 'two numbers are not equal' - Not P: 'the numbers are equal'.

Step 4: Formulate the Contrapositive

Using the identification in Step 3, the contrapositive of the given statement is: 'If the squares of two numbers are equal, then the numbers are equal'.

Step 5: Match with Given Options

Among the options provided, option (a) matches the contrapositive derived in Step 4: 'If the squares of two numbers are equal, then the numbers are equal'.

Key Concepts

Contrapositive Statements

In mathematical logic, understanding contrapositive statements is an essential skill. A contrapositive is a type of logical statement that changes the meaning of the original conditional statement while maintaining its truth.
Let's break down what a contrapositive means. Whenever you have a conditional statement of the form "If P, then Q," the contrapositive flips this around to form "If not Q, then not P." For instance, consider the conditional statement: "If two numbers are not equal, then their squares are not equal." In this context:

• P is "two numbers are not equal,"
• Q is "their squares are not equal."

The contrapositive of this statement would be "If the squares of two numbers are equal, then the numbers are equal."
It is important to note that a contrapositive is always logically equivalent to its original statement, meaning that if the original statement is true, so is its contrapositive.

Conditional Statements

Conditional statements are foundational in logic and mathematics, often taking the form "If P, then Q." These statements express a logical relationship between an antecedent (P) and a consequent (Q).
For example, our statement "If two numbers are not equal, then their squares are not equal" is a conditional statement where:

• The antecedent, P, is "two numbers are not equal,"
• The consequent, Q, is "their squares are not equal."

Understanding the structure of conditional statements helps in forming contrapositives, inversions, and converses—important logical tools used to analyze statements in mathematics.
Besides, recognizing these components allows for effective problem-solving and accurate interpretation of logical arguments.

Logical Equivalence

Logical equivalence is a concept in logic that describes two statements that always have the same truth value. When two statements are logically equivalent, they either both are true, or both are false under the same conditions.
In the context of conditional statements and their contrapositives, logical equivalence plays a significant role. A conditional statement "If P, then Q" is always logically equivalent to its contrapositive "If not Q, then not P." For example, as shown in the original exercise, the statement "If two numbers are not equal, then their squares are not equal" is logically equivalent to its contrapositive "If the squares of two numbers are equal, then the numbers are equal."
This logical equivalence means that you can trust the contrapositive of a true statement to be true as well. It is a powerful tool in mathematical proofs and problem-solving, ensuring that you can draw reliable conclusions across different logical formulations of the same idea.