1. Chapter 4 Numeration and Mathematical Systems © 2008 Pearson Addison-Wesley. All rights reserved

2. © 2008 Pearson Addison-Wesley. All rights reserved 4-5-2 Chapter 4: Numeration and Mathematical Systems 4.1 Historical Numeration Systems 4.2 Arithmetic in the Hindu-Arabic System 4.3 Conversion Between Number Bases 4.4 Clock Arithmetic and Modular Systems 4.5 Properties of Mathematical Systems 4.6Groups

3. © 2008 Pearson Addison-Wesley. All rights reserved 4-5-3 Chapter 1 Section 4-5 Properties of Mathematical Systems

4. © 2008 Pearson Addison-Wesley. All rights reserved 4-5-4 Properties of Mathematical Systems An Abstract System Closure Property Commutative Property Associative Property Identity Property Inverse Property Distributive Property

5. © 2008 Pearson Addison-Wesley. All rights reserved 4-5-5 An Abstract System The focus will be on elements and operations that have no implied mathematical significance. We can investigate the properties of the system without notions of what they might be.

6. © 2008 Pearson Addison-Wesley. All rights reserved 4-5-6 Operation Table Consider the mathematical system with elements {a, b, c, d} and an operation denoted by ☺. The operation table on the next slide shows how operation ☺ combines any two elements. To use the table to find c ☺ d, locate c on the left and d on the top. The row and column intersect at b, so c ☺ d = b.

7. © 2008 Pearson Addison-Wesley. All rights reserved 4-5-7 Operation Table for ☺ ☺abcd aabcd bbdac ccadb ddcba

8. © 2008 Pearson Addison-Wesley. All rights reserved 4-5-8 Closure Property For a system to be closed under an operation, the answer to any possible combination of elements from the system must in the set of elements. ☺abcd aabcd bbdac ccadb ddcba This system is closed.

9. © 2008 Pearson Addison-Wesley. All rights reserved 4-5-9 Commutative Property For a system to have the commutative property, it must be true that for any elements X and Y from the set, X ☺ Y = Y ☺ X. ☺abcd aabcd bbdac ccadb ddcba This system has the commutative property. The symmetry with respect to the diagonal line shows this property

10. © 2008 Pearson Addison-Wesley. All rights reserved 4-5-10 Associative Property For a system to have the associative property, it must be true that for any elements X, Y, and Z from the set, X ☺ (Y ☺ Z) = (X ☺ Y) ☺ Z. ☺abcd aabcd bbdac ccadb ddcba This system has the associative property. There is no quick check – just work through cases.

11. © 2008 Pearson Addison-Wesley. All rights reserved 4-5-11 Identity Property For the identity property to hold, there must be an element E in the set such that any element X in the set, X ☺ E = X and E ☺ X = X. ☺abcd aabcd bbdac ccadb ddcba a is the identity element of the set.

12. © 2008 Pearson Addison-Wesley. All rights reserved 4-5-12 Inverse Property If there is an inverse in the system then for any element X in the system there is an element Y (the inverse of X) in the system such that X ☺ Y = E and Y ☺ X = E, where E is the identity element of the set. You can inspect the table to see that every element has an inverse. ☺abcd aabcd bbdac ccadb ddcba

13. © 2008 Pearson Addison-Wesley. All rights reserved 4-5-13 Potential Properties of a Single Operation Symbol Let a, b, and c be elements from the set of any system, and ◘ represent the operation of the system. Closure a ◘ b is in the set Inverse there exists an element x in the set such that a ◘ x = e and x ◘ a = e. Commutative a ◘ b = a ◘ b. Identity The system has an element e such that a ◘ e = a and e ◘ a = a. Associative a ◘ (b ◘ c) = (a ◘ b) ◘ c

14. © 2008 Pearson Addison-Wesley. All rights reserved 4-5-14 Example: Identifying Properties Consider the system shown with elements {0, 1, 2, 3, 4} and operation Which properties are satisfied by this system? 01234 000000 101234 202402 303030 404204

15. © 2008 Pearson Addison-Wesley. All rights reserved 4-5-15 Example: Identifying Properties Solution The system satisfies the closure, associative, commutative, and identity properties, but not the inverse property. 01234 000000 101234 202402 303030 404204

16. © 2008 Pearson Addison-Wesley. All rights reserved 4-5-16 Distributive Property Let ☺ and ◘ be two operations defined for elements in the same set. Then ☺ is distributive over ◘ if a ☺ (b ◘ c) = (a ☺ b) ◘ (a ☺ c) for every choice of elements a, b, and c from the set.

17. © 2008 Pearson Addison-Wesley. All rights reserved 4-5-17 Example: Testing for the Distributive Property Is addition distributive over multiplication on the set of whole numbers? Solution We check the statement below: Notice, it fails when using 1, 2, and 3: This counterexample shows that addition is not distributive over multiplication.