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SEVENTH EDITION and EXPANDED SEVENTH EDITION

SEVENTH EDITION and EXPANDED SEVENTH EDITION. Chapter 10. Mathematical Systems. 10.1. Groups. Definitions. A mathematical system consists of a set of elements and at least one binary operation.

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SEVENTH EDITION and EXPANDED SEVENTH EDITION

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  1. SEVENTH EDITION and EXPANDED SEVENTH EDITION

  2. Chapter 10 Mathematical Systems

  3. 10.1 Groups

  4. Definitions • A mathematical system consists of a set of elements and at least one binary operation. • A binary operation is an operation, or rule, that can be performed on two and only two elements of a set.

  5. For elements a, b, and c Addition Multiplication Commutative property a + b = b + a ab = ba Associate property (a + b) + c = a + (b + c) (ab)c = a(bc) Properties

  6. Closure • If a binary operation is performed on any two elements of a set and the result is an element of the set, then that set is closed (or has closure) under the given binary operation.

  7. Identity Element • An identity element is an element in a set such that when a binary operation is performed on it and any given element in the set, the result is the given element. • Additive identity element is 0. • Multiplicative identity element is 1.

  8. Inverses • When a binary operation is performed on two elements in a set and the result is the identity element for the binary operation, each element is said to be the inverse of the other.

  9. Properties of a Group • Any mathematical system that meets the following four requirements is called a group. • The set of elements is closed under the given operation. • An identity element exists for the set under the given operation. • Every element in the set has an inverse under the given operation. • The set of elements is associative under the given operation.

  10. Commutative Group • A group that satisfies the commutative property is called a commutative group or (abelian group).

  11. Properties of a Commutative Group • A mathematical system is a commutative group if all five conditions hold. • The set of elements is closed under the given operation. • An identity element exists for the set under the given operation. • Every element in the set has an inverse under the given operation. • The set of elements is associative under the given operation. • The set of elements is commutative under the given conditions.

  12. 10.2 Finite Mathematical Systems

  13. Definition • A finite mathematical system is one whose set contains a finite number of elements. • Example: Determine whether the clock arithmetic system under the operation of addition is a commutative group.

  14. Definition continued • Closure: The set of elements in clock arithmetic is closed under the operation of addition. • Identity: There is an additive identity element, namely 12. • Inverse elements: Each element in the set has an inverse. • Associative property: The system is associative under the operation of addition. • Commutative property: The commutative property of addition is true for clock arithmetic. • The system satisfies the five properties required for a mathematical system. Thus, clock arithmetic under the operation of addition is a commutative or abelian group.

  15. 10.3 Modular Arithmetic

  16. Definition • A modulo m system consists of m elements, 0 through m 1, and a binary operation. • a is congruent to b modulo m, written a  b(mod m), if a and b have the same remainder when divided by m.

  17. Example • Determine which number from 0 to 7, the following numbers are congruent to in modulo 8. • a) 66 b) 72 c) 109

  18.  remainder Solution: a) 66 • To determine the value 66 is congruent to in mod 8, divide 66 by 8 and find the remainder. Thus, 66  2 (mod 8)

  19. b) 72 72  ? (mod 8) 72  8 = 9, remainder 0 72  0 (mod 8) c) 109 109  ? (mod 8) 109  8 = 13, remainder 5 109  5 (mod 8) Solutions continued

  20. Example • Evaluate each in mod 6. • a) 4 + 2 b) 3  2 c) 2(4) • a) 4 + 2  ? (mod 6) 6  ? (mod 6) 6  6 = 1, remainder 0. Therefore, 4 + 2  0 (mod 6)

  21. b) 3  2 3  2  ? (mod 6) 1  ? (mod 6) 1  1 (mod 6) c) 4(2) 4(2)  ? (mod 6) 8  ? (mod 6) 8  6 = 1, remainder 2. Therefore, 4(2)  2 (mod 6) Example continued

  22. a) 4 • ?  3(mod 5) One method is to replace the ? mark with the numbers. 4 • 0  0(mod 5) 4 • 1  4(mod 5) 4 • 2  3(mod 5) 4 • 3  2(mod 5) 4 • 4  1(mod 5) Therefore, ? = 2 since 4 • 2  3(mod 5). b) 3 • ?  0(mod 5) 3 • 0  0(mod 5) 3 • 1  3(mod 5) 3 • 2  0(mod 5) 3 • 3  3(mod 5) 3 • 4  0(mod 5) 3 • 5  3(mod 5) Therefore, the 0, 2 and 4 result in true statements. Find all replacements for the question mark that make the statements true.

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