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10. Suppose x, y, ze Z and x #0. If xyz, then xty and xtz. 11. Suppose x,y e Z. If x²(y + 3) is even, then x is even or y is odd. 12. Suppose a e Z. If a² is not divisible by 4, then a is odd. 13. Suppose xe R. If x5 +7x³ +5x2x4+x² +8, then x ≥ 0.

10. Suppose x, y, ze Z and x #0. If xyz, then xty and xtz. 11. Suppose x,y e Z. If x²(y + 3) is even, then x is even or y is odd. 12. Suppose a e Z. If a² is not divisible by 4, then a is odd. 13. Suppose xe R. If x5 +7x³ +5x2x4+x² +8, then x ≥ 0.

Advanced Engineering Mathematics
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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Please a detailed prove would be appreciated for number 10, 12 13
**Exercises for Chapter 5** **A. Prove the following statements with contrapositive proof.** (In each case, think about how a direct proof would work. In most cases contrapositive is easier.) 1. Suppose \( n \in \mathbb{Z} \). If \( n^2 \) is even, then \( n \) is even. 2. Suppose \( n \in \mathbb{Z} \). If \( n^2 \) is odd, then \( n \) is odd. 3. Suppose \( a, b \in \mathbb{Z} \). If \( a^2(b^2 - 2b) \) is odd, then \( a \) and \( b \) are odd. 4. Suppose \( a, b, c \in \mathbb{Z} \). If \( a \) does not divide \( bc \), then \( a \) does not divide \( b \). 5. Suppose \( x \in \mathbb{R} \). If \( x^2 + 5x < 0 \) then \( x < 0 \). 6. Suppose \( x \in \mathbb{R} \). If \( x^3 - x > 0 \) then \( x > -1 \). 7. Suppose \( a, b \in \mathbb{Z} \). If both \( ab \) and \( a + b \) are even, then both \( a \) and \( b \) are even. 8. Suppose \( x \in \mathbb{R} \). If \( x^5 - 4x^4 + 3x^3 - x^2 + 3x - 4 \geq 0 \), then \( x \geq 0 \). 9. Suppose \( n \in \mathbb{Z} \). If \( 3 \mid n^2 \), then \( 3 \mid n \). 10. Suppose \( x, y, z \in \mathbb{Z} \) and \( x \neq 0 \). If \( x \nmid yz \), then \( x \nmid y \) and \( x \nmid z \). 11. Suppose \( x, y \in \mathbb{Z} \). If \( x^2(y + 3) \)
Transcribed Image Text:**Exercises for Chapter 5** **A. Prove the following statements with contrapositive proof.** (In each case, think about how a direct proof would work. In most cases contrapositive is easier.) 1. Suppose \( n \in \mathbb{Z} \). If \( n^2 \) is even, then \( n \) is even. 2. Suppose \( n \in \mathbb{Z} \). If \( n^2 \) is odd, then \( n \) is odd. 3. Suppose \( a, b \in \mathbb{Z} \). If \( a^2(b^2 - 2b) \) is odd, then \( a \) and \( b \) are odd. 4. Suppose \( a, b, c \in \mathbb{Z} \). If \( a \) does not divide \( bc \), then \( a \) does not divide \( b \). 5. Suppose \( x \in \mathbb{R} \). If \( x^2 + 5x < 0 \) then \( x < 0 \). 6. Suppose \( x \in \mathbb{R} \). If \( x^3 - x > 0 \) then \( x > -1 \). 7. Suppose \( a, b \in \mathbb{Z} \). If both \( ab \) and \( a + b \) are even, then both \( a \) and \( b \) are even. 8. Suppose \( x \in \mathbb{R} \). If \( x^5 - 4x^4 + 3x^3 - x^2 + 3x - 4 \geq 0 \), then \( x \geq 0 \). 9. Suppose \( n \in \mathbb{Z} \). If \( 3 \mid n^2 \), then \( 3 \mid n \). 10. Suppose \( x, y, z \in \mathbb{Z} \) and \( x \neq 0 \). If \( x \nmid yz \), then \( x \nmid y \) and \( x \nmid z \). 11. Suppose \( x, y \in \mathbb{Z} \). If \( x^2(y + 3) \)
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