**Mathematical Propositions and Proofs****Topic: Proof by Contraposition****The Statement:**"Give a proof by contraposition of the following statement."- **If $ x $ is an integer and the square of $ x $ is even, then $ x $ is even.****Concept Overview:**In mathematics, a proof by contraposition is a powerful technique used to prove statements of the form "If $ P $, then $ Q $." Instead of proving this directly, we can equivalently prove the contrapositive statement: "If not $ Q $, then not $ P $."**Application to the Given Statement:****Original Statement:** If $ x $ is an integer and the square of $ x $ is even, then $ x $ is even.**Contrapositive Statement:** If $ x $ is not even (i.e., $ x $ is odd), then the square of $ x $ is not even (i.e., the square of $ x $ is odd).**Proof by Contraposition:**1. **Start with the assumption that $ x $ is odd.** - An integer $ x $ is odd if it can be written in the form: $ x = 2k + 1 $, where $ k $ is an integer.2. **Square both sides to find the square of $ x $:** - $ x^2 = (2k + 1)^2 $ - $ x^2 = 4k^2 + 4k + 1 $ - $ x^2 = 2(2k^2 + 2k) + 1 $3. **Analyze the resulting expression:** - The expression $ 2(2k^2 + 2k) $ is clearly even since it is multiplied by 2. - Adding 1 to an even number yields an odd number.4. **Conclude that the square of $ x $ is odd:** - Therefore, $ x^2 $ is odd when $ x $ is odd.By proving the contrapositive statement, we have effectively proven the original statement: If $ x $ is an integer and the square of $ x $ is even, then $ x $ is even.This concludes the

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Description: **Mathematical Propositions and Proofs****Topic: Proof by Contraposition****The Statement:**"Give a proof by contraposition of the following statement."- **If $ x $ is an integer and the square of $ x $ is even, then $ x $ is even.****Concept

To do this, we must first state the contrapositive of what we set out to prove. so If x2 is even…

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Give a proof by contraposition of the following statement. If x is an integer and square of x is even, then x is even.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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**Mathematical Propositions and Proofs**

**Topic: Proof by Contraposition**

**The Statement:**
"Give a proof by contraposition of the following statement."

- **If $ x $ is an integer and the square of $ x $ is even, then $ x $ is even.**

**Concept Overview:**
In mathematics, a proof by contraposition is a powerful technique used to prove statements of the form "If $ P $, then $ Q $." Instead of proving this directly, we can equivalently prove the contrapositive statement: "If not $ Q $, then not $ P $."

**Application to the Given Statement:**

**Original Statement:**
If $ x $ is an integer and the square of $ x $ is even, then $ x $ is even.

**Contrapositive Statement:**
If $ x $ is not even (i.e., $ x $ is odd), then the square of $ x $ is not even (i.e., the square of $ x $ is odd).

**Proof by Contraposition:**

1. **Start with the assumption that $ x $ is odd.**
- An integer $ x $ is odd if it can be written in the form: $ x = 2k + 1 $, where $ k $ is an integer.

2. **Square both sides to find the square of $ x $:**
- $ x^2 = (2k + 1)^2 $
- $ x^2 = 4k^2 + 4k + 1 $
- $ x^2 = 2(2k^2 + 2k) + 1 $

3. **Analyze the resulting expression:**
- The expression $ 2(2k^2 + 2k) $ is clearly even since it is multiplied by 2.
- Adding 1 to an even number yields an odd number.

4. **Conclude that the square of $ x $ is odd:**
- Therefore, $ x^2 $ is odd when $ x $ is odd.

By proving the contrapositive statement, we have effectively proven the original statement: If $ x $ is an integer and the square of $ x $ is even, then $ x $ is even.

This concludes the

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