CHAPTER 9 Review Exercises 581 Important Facts and Concepts Examples and Discussion Section 9.1, 9.2 A mathematical system consists of a set of elements and at least one binary operation. Commutative Property Addition: a b b a + = + Multiplication: a b b a ⋅ = ⋅ Associative Property Addition: a b c a b c ( ) ( ) + + = + + Multiplication: a b c a b c ( ) ( ) ⋅ ⋅ = ⋅ ⋅ Properties of a Group and a Commutative Group A mathematical system is a group if the first four following conditions hold, and a commutative group if all five conditions hold. 1. The set of elements is closed under the given operation. 2. An identity element exists for the set. 3. Every element in the set has an inverse. 4. The set of elements is associative under the given operation. 5. The set of elements is commutative under the given operation. Discussion, page 542 Discussion, pages 542–543 Discussion, pages 542–543 Discussion, pages 543–546 Examples 1–3, pages 546–547, Examples 1– 6, pages 551–555, Example 8, pages 575–577 If the elements of a finite mathematical system are symmetric about the main diagonal, then the system is commutative. Discussion, page 552, Examples 2– 6, pages 553–556, Section 9.3 a is congruent to b modulo m, written a b mod m ( ), ≡ if a and b have the same remainder when divided by m. Discussion, pages 560–561, Examples 1– 6, pages 560–566 Section 9.4 A matrix is a rectangular array of real numbers, symbols, or expressions. Matrix Operations Addition and Subtraction Multiplication by a Scalar Multiplication Matrices as a Commutative Group See Section 9.1 for the five properties of a commutative group. Discussion, pages 569–570 Examples 1–3, pages 570–571, Example 4, pages 571–572, Examples 5–7, pages 572–575 Example 8, pages 575–577 Summary CHAPTER 9 9.1, 9.2 1. What is a binary operation? A binary operation is an operation that can be performed on two and only two elements of a set. The result is a single element. 2. List the parts of a mathematical system. A mathematical system consists of a set of elements and at least one binary operation. Review Exercises CHAPTER 9 3. Is the set of whole numbers closed under the operation of division? Explain. No; for example, 2 3 2 3 and 2 3 ÷ = is not a whole number. 4. Is the set of real numbers closed under the operation of subtraction? Explain. Yes; the difference of any two real numbers is a real number.
RkJQdWJsaXNoZXIy NjM5ODQ=