566 CHAPTER 9 Mathematical Systems Did You Know? Secrets of the Universe m A computer simulation of protons and neutrons being smashed together. Physicists use group theory in their search for the elementary particles that make up all matter. The operations are transformations: the coming together or splitting apart of elementary particles as they form larger particles such as protons and neutrons. Some transformations bring two particles together to form a different particle (binary operation), some leave a particle intact (identity), and some follow the rules of associativity and commutativity. In fact, understanding there were distinct “families” of particles related to one another as groups enabled physicists to predict the existence of certain particles before any experimental evidence was available. 3. Inverse elements : Does every element have an inverse? Recall that an element plus its inverse must equal the identity element. In this example, the identity element is 0. Therefore, for each of the given elements 0, 1, 2, 3, and 4, we must find the element that when added to it results in a sum of zero. These elements will be the corresponding inverses. + = + = + = + = + = + = Element Inverse Identity 0 ? 0 1 ? 0 2 ? 0 3 ? 0 4 ? 0 + ≡ Since 0 0 0, 0 is its own inverse. + ≡ Since 1 4 0, 4 is the inverse of 1. Since 2 3 0, 3 is the inverse of 2. + ≡ Since 3 2 0, 2 is the inverse of 3. + ≡ + ≡ Since 4 1 0, 1 is the inverse of 4. Note that each element has an inverse. 4. Associative property : It is given that the associative property holds. One example that illustrates the associative property is + + = + + + = + = (23)4 2(34) 0 4 2 2 4 4 ? ? True 5. Commutative property : Is a b b a + = + for all elements a and b of the given set? The table shows that the system is commutative because the elements are symmetric about the main diagonal. We will give one example to illustrate the commutative property. + = + = 4 2 2 4 1 1 ? True All five properties are satisfied. Thus, modulo 5 arithmetic under the operation of addition is a commutative group. 7 Now try Exercise 65 Instructor Resources for Section 9.3 in MyLab Math • Objective-Level Videos 9.3 • PowerPoint Lecture Slides 9.3 • MyLab Exercises and Assignments 9.3 Exercises Warm Up Exercises In Exercises 1– 6, fill in the blanks with an appropriate word, phrase, or symbol(s). 1. The elements in a modulo m system consist of the m numbers, 0 through _______. m 1. − 2. In a modulo m system, in addition to the m elements, there is also a(n) _______ operation. Binary 3. Whenever a process is repetitive, _______ arithmetic may be helpful in answering some questions about the process. Modular 4. In a modulo m system, the number m is called the _______ of the system. Modulus 5. If a and b have the same remainder when divided by m, we say that a is _______ to b modulo m. Congruent 6. To determine the value that 45 is congruent to in modulo 7, we _______ 45 by 7 and determine the remainder. Divide Practice the Skills In Exercises 7–16, determine what number the sum, difference, or product is congruent to in modulo 5 . SECTION 9.3 PopaDuhu/Shutterstock.
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