Problem 43 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Consider the statement: "For an integer \(\mathrm{n}\), if \(\mathrm{n}^{3}-1\) is even, then \(\mathrm{n}\) is odd." The contrapositive statement of this statement is: \(\quad\) (a) For an integer \(\mathrm{n}\), if \(\mathrm{n}\) is even, then \(\mathrm{n}^{3}-1\) is odd. (b) For an intetger \(\mathrm{n}\), if \(\mathrm{n}^{3}-1\) is not even, then \(\mathrm{n}\) is not odd. (c) For an integer \(\mathrm{n}\), if \(\mathrm{n}\) is even, then \(\mathrm{n}^{3}-1\) is even. (d) For an integer \(\mathrm{n}\), if \(\mathrm{n}\) is odd, then \(\mathrm{n}^{3}-1\) is even.

Step-by-Step Solution

Answer

The contrapositive is option (a).

Step 1: Understand the Statement

The given statement is "For an integer \( n \), if \( n^3 - 1 \) is even, then \( n \) is odd." This is a conditional statement, often written as "if \( P \), then \( Q \)", where here \( P \) is "\( n^3 - 1 \) is even" and \( Q \) is "\( n \) is odd".

Step 2: Write the Contrapositive

The contrapositive of a statement "if \( P \), then \( Q \)" is "if not \( Q \), then not \( P \)". So, for our statement, the contrapositive is "if \( n \) is not odd, then \( n^3 - 1 \) is not even."

Step 3: Simplify the Contrapositive

Recognize that "\( n \) is not odd" is the same as saying "\( n \) is even." Likewise, "\( n^3 - 1 \) is not even" means "\( n^3 - 1 \) is odd." Thus, the contrapositive is simplified to: "For an integer \( n \), if \( n \) is even, then \( n^3 - 1 \) is odd."

Step 4: Match to Options

Compare the simplified contrapositive to the given options. The contrapositive "if \( n \) is even, then \( n^3 - 1 \) is odd" matches option (a).

Key Concepts

Conditional Statements

Conditional statements are an essential concept in mathematical logic. They enable us to make conclusions about one condition based on the existence of another. A conditional statement typically takes the form "if \( P \), then \( Q \)", where \( P \) is the hypothesis, and \( Q \) is the conclusion. These statements help us establish relationships between different mathematical conditions and make logical deductions.

• "If \( P \)" introduces the condition assumed to be true.
• "Then \( Q \)" indicates the result that follows under this assumption.

In this exercise, we're given the statement: "For an integer \( n \), if \( n^3 - 1 \) is even, then \( n \) is odd." The hypothesis is that \( n^3 - 1 \) is even, and the conclusion is that \( n \) is odd. Writing the contrapositive changes the original conditional statement into another form that is logically equivalent. In logical terms, the contrapositive of a statement "if \( P \), then \( Q \)" becomes "if not \( Q \), then not \( P \)".

Mathematical Logic

Mathematical logic provides the foundation for establishing sound mathematical truths. It involves the process of forming compound statements from simple ones, as well as rules for determining the truth or falsity of these statements.
To understand mathematical logic, it's crucial to grasp the concept of logical equivalence, where different expressions can represent the same reality. In the exercise, the original statement and its contrapositive are considered logically equivalent. This means if one is true, the other must be true as well.

• The process of finding a contrapositive involves reversing and negating the hypothesis and conclusion of the original statement.
• Logical equivalence allows mathematicians to use different perspectives to prove a truth.

Grasping these logic principles is valuable for learning how mathematical proofs are structured and verified. It shows how altering statements while preserving truth can provide alternate means of reaching the same conclusion.

Even and Odd Numbers

Understanding the properties of even and odd numbers is fundamental to solving many mathematical problems. Even numbers are integers that can be divided by 2 without a remainder, such as 2, 4, and 6, whereas odd numbers produce a remainder of 1 when divided by 2, like 1, 3, and 5.
In this exercise, the properties of even and odd numbers help us deduce the results of operations like \( n^3 - 1 \).

• For any integer \( n \), if \( n \) is even, then \( n^3 - 1 \) differs from an even product, producing an odd number (as the subtraction of 1 turns the parity).
• Conversely, if \( n \) were odd, \( n^3 - 1 \) would be even, as odd cubed still results in an odd integer minus 1.

This understanding underlines the simplicity yet profundity of number properties in mathematical proofs, helping offer elegant solutions to seemingly complex problems through basic arithmetic principles.