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6. Suppose xER. If x³. -x>0 then x>-1. 7. Suppose a, b e Z. If both ab and a + b are even, then both a and b are even. 8. Suppose x E R. If x5 - 4x4 + 3x³ - x² + 3x-4 ≥0, then x ≥ 0.

6. Suppose xER. If x³. -x>0 then x>-1. 7. Suppose a, b e Z. If both ab and a + b are even, then both a and b are even. 8. Suppose x E R. If x5 - 4x4 + 3x³ - x² + 3x-4 ≥0, then x ≥ 0.

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Number 6,7,8,10,12,13
**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 ∤ \( n^2 \), then 3 ∤ \( n \). 10. Suppose \( x, y, z \in \mathbb{Z} \) and \( x \neq 0 \). If \( x \mid yz \), 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 ∤ \( n^2 \), then 3 ∤ \( n \). 10. Suppose \( x, y, z \in \mathbb{Z} \) and \( x \neq 0 \). If \( x \mid yz \), then \( x \nmid y \) and \( x \nmid z \). 11. Suppose \( x, y \in \mathbb{Z} \). If \( x^2(y + 3) \)
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