542 CHAPTER 9 Mathematical Systems Mathematical Systems We begin our discussion by introducing a mathematical system. As you will learn shortly, you already know and use many mathematical systems. How big is the universe? Is there an end to the universe? Does the universe continue to expand? Such questions are addressed by the scientific theory known as the big bang theory. In this section, we will introduce a mathematical structure known as a group . Many modern scientific theories, including the big bang theory, make use of groups and other mathematical topics discussed in this book. Groups SECTION 9.1 LEARNING GOALS Upon completion of this section, you will be able to: 7 Understand and identify mathematical systems. 7 Understand and use the commutative and associative properties. 7 Understand and identify closure. 7 Understand and identify identity elements and inverse elements. 7 Show whether a mathematical system is a group or a commutative group. Why This Is Important The more we understand about our universe, the better prepared we are to answer questions regarding the history and future of planet Earth. Definition: Mathematical System A mathematical system consists of a set of elements and at least one binary operation. In the above definition, we mention binary operation. A binary operation is an operation, or rule, that can be performed on two and only two elements of a set. The result is a single element. When we add two integers, the sum is one integer. When we multiply two integers, the product is one integer. Thus, addition and multiplication are both binary operations. Is determining the reciprocal of a number a binary operation? No, it is an operation on a single element of a set. Timely Tip Throughout this section, we will work with many different sets of numbers, such as natural numbers, whole numbers, integers, rational numbers, and real numbers. A review of the numbers that make up each of these sets can be found in Section 5.5. When you learned how to add integers, you were introduced to a mathematical system. The set of elements is the set of integers, and the binary operation is addition. When you learned how to multiply integers, you became familiar with a second mathematical system. The set of integers with the operation of subtraction and the set of integers with the operation of division are two other examples of mathematical systems, since subtraction and division are also binary operations. Some mathematical systems are used in solving everyday problems, such as planning work schedules. Others are more abstract and are used primarily in research, chemistry, physical structure, matter, the nature of genes, and other scientific fields. Commutative and Associative Properties Once a mathematical system is defined, its structure may display certain properties. Consider the set of integers { } = − − − I . . . , 3, 2, 1, 0, 1, 2, 3, . . . The set of integers can be studied with the operations of addition, subtraction, multiplication, and division as separate mathematical systems. For example, when we study the set of integers under the operations of addition or multiplication, we see that the commutative and associative properties hold. The general forms of these properties are shown in the following box. Irina Dmitrienko/ Alamy Stock Photo
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