Database System Concepts
Database System Concepts
7th Edition
ISBN: 9780078022159
Author: Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher: McGraw-Hill Education
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Concept explainers

Operations
In mathematics and computer science, an operation is an event that is carried out to satisfy a given task. Basic operations of a computer system are input, processing, output, storage, and control.
Basic Operators
An operator is a symbol that indicates an operation to be performed. We are familiar with operators in mathematics; operators used in computer programming are—in many ways—similar to mathematical operators.
Division Operator
We all learnt about division—and the division operator—in school. You probably know of both these symbols as representing division:
Modulus Operator
Modulus can be represented either as (mod or modulo) in computing operation. Modulus comes under arithmetic operations. Any number or variable which produces absolute value is modulus functionality. Magnitude of any function is totally changed by modulo …
Operators
In the realm of programming, operators refer to the symbols that perform some function. They are tasked with instructing the compiler on the type of action that needs to be performed on the values passed as operands. Operators can be used in mathematical…
Question
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 \( x^2(y + 3) \)
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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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