1. © 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 13 Mathematical Systems

2. © 2010 Pearson Prentice Hall. All rights reserved. 2 13.1 Mathematical Systems

3. © 2010 Pearson Prentice Hall. All rights reserved. Objectives 1.Understand what is meant by a mathematical system. 2.Understand properties of certain mathematical systems. 3

4. © 2010 Pearson Prentice Hall. All rights reserved. Mathematical Systems A binary operation is a rule that can be used to combine any two elements of a set, resulting in a single element. A mathematical system consists of a set of elements and at least one binary operation. 4

5. © 2010 Pearson Prentice Hall. All rights reserved. Example 1: An Example of a Mathematical System If two even or odd numbers are added, the sum is even. For example, 4 + 4 = 8 (even + even = even) or 3 + 3 = 6 (odd + odd = even). Let E represent any even number and O any odd number. a.What is the set of elements of this mathematical system? b.What is the binary operation of this mathematical system? c.What is E + O ? 5

6. © 2010 Pearson Prentice Hall. All rights reserved. Example 1 continued Solution: a.The set of elements of this mathematical system is {E, O}. b.The binary operation used to combine any two elements of {E, O} is +. c.We find E + O by finding the intersection of row and column O. So, E + O = O. 6

7. © 2010 Pearson Prentice Hall. All rights reserved. Properties of Some Mathematical Systems Suppose a binary operation is performed on any two elements of a set. If the result is an element of the set, then that set is closed (or has closure) under the binary operation. 7

8. © 2010 Pearson Prentice Hall. All rights reserved. Example 4: Understanding the Closure Property Is the set of integers closed under the operation of division? Solution: If any two integers are divided, the quotient might not be an integer. For example, Thus, the set of integers is not closed under division. 8

9. © 2010 Pearson Prentice Hall. All rights reserved. Properties of Some Mathematical Systems Suppose that represents a binary operation for the elements of a set. The set is commutative under the operation if for any two elements of the set a and b, The order of the two elements can be switched without changing the answer. 9

10. © 2010 Pearson Prentice Hall. All rights reserved. The Commutative Property We use the table to show that. Solution: Observe is found by locating 2 on the left and then 3 across the top. The intersection shows. Next, we find by locating 3 on the left and 2 across the top. The intersection shows. Thus, because both binary operations give 1. 10

11. © 2010 Pearson Prentice Hall. All rights reserved. The Commutative Property Drawing a diagonal line from the upper left corner to the lower right corner gives a mirror for the part below the diagonal. Thus, the set {0, 1, 2, 3} is commutative under the operation. 11

12. © 2010 Pearson Prentice Hall. All rights reserved. Properties of Some Mathematical Systems A set of elements is associative under a given operation if for any three elements of the set, If a binary operation is associative, the answer does not change if we group the first two elements together or the last two elements together. 12

13. © 2010 Pearson Prentice Hall. All rights reserved. Example 5: Verifying One Case for the Associative Property Use the table to verify the associative property for a = 3, b = 2, and c = 1. 13

14. © 2010 Pearson Prentice Hall. All rights reserved. Example 5 continued Solution: Changing the location of the parenthesis does not change the answer, 2. 14

15. © 2010 Pearson Prentice Hall. All rights reserved. The Identity Property We call 0 the identity element of addition and 1 the identity element of multiplication. In a mathematical system, the identity element (if there is one) is an element from the set such that when a binary operation is performed on any element in the set and the identity element, the result is the given element. 15

16. © 2010 Pearson Prentice Hall. All rights reserved. The Inverse Property If a binary operation is performed on two elements in a set and the result is the identity element, then each element is called the inverse of each other. Example: 16