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Section 10.1 Groups Guided Notes

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Section 10.1 Groups Guided Notes Section 10.1 Groups Guided Notes What You Will Learn Mathematical Systems Commutative Property Associative Property Closure Identity Element Inverses Groups Let’s begin with, a Mathematical System: A mathematical system consists of a set of elements and at least one binary operation. 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. Addition, multiplication, subtraction and division are all binary operations. The set of integers and the binary operation of addition make a mathematical system. Commutative Property For any elements a, b, and c Addition a + b = b + a Multiplication a • b = b • a Associative Property For any elements a, b, and c Addition (a + b) + c = a + (b + c) Multiplication (a • b) • c = a • (b • c) 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. 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. 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. The additive inverse of a nonzero integer, a, is –a. 0 is its own additive inverse. The multiplicative inverse of a is 1/a. However, it may not be an integer; so most integers do not have a multiplicative inverse in the set of integers.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. Commutative Group A group that satisfies the commutative property is called a commutative group (or abelian group). 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 operations. Example : Whole Numbers Under Addition Determine whether the mathematical system consisting of the set of whole numbers under the operation of addition forms a group. Solution 1. Closure: Sum of any two whole numbers is a whole number. The set is closed under addition. 2. Identity element: 0 is the additive identity. a + 0 = a. The system contains an identity element. 3. Inverse elements: The additive inverse is the opposite of the number. –1 is the additive inverse of 1, – 2 is that of 2, and so on. The numbers, –1, –2, … are not in the set of whole numbers. The mathematical system is NOT a group. Example : Real Numbers Under Addition Determine whether the set of real numbers under the operation of addition forms a commutative group. Solution 1. Closure: Sum of any two real numbers is a whole number. The set is closed under addition. 2. Identity element: 0 is the additive identity. a + 0 = a + 0 = a. The system contains an identity element. 3. Inverse elements: The additive inverse is the opposite of the number. –1 is the additive inverse of 1, and –a is that of a. a +(–a) = –a + a = 0. All real numbers have an additive inverse. 4. Associative property: for any real numbers a, b and c, (a + b) + c = a + (b + c). 5. Commutative property: for any real numbers a and b, a + b = b + a. The set of real numbers forms a commutative group. YOUR TURN!------------------------------- Section 10.2 Finite Mathematical Systems What You Will Learn Finite Mathematical Systems: Clock Arithmetic Mathematical Systems without Numbers Definition A finite mathematical system is one whose set contains a finite number of elements. Clock 12 Arithmetic The set of elements is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. The binary operation we will use is addition, a movement clockwise. For example, 4 + 9 = 1. Also, 9 + 4 = 1. An addition table is started DEMO-. Example: Determine whether the clock 12 arithmetic system under the operation of addition is a commutative group. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} Solution 1. Closure: Note that the table contains only elements of the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. The set is closed under addition. 2. Identity element: 12 is the additive identity. 4 + 12 = 12 + 4 = 4 3. Inverse elements: The additive inverse is the number that when added will yield the additive identity, 12. 8 is the additive identity of 4; 4 + 8 = 12. And 4 is the additive identity of 8; 8 + 4 = 12. Similarly, 7 and 5 are additive identities of each other. Here is the table of additive identities: 4. Associative property: for all values of a, b and c, does (a + b) + c = a + (b + c)? Let a = 2, b = 6, and c = 8. (2 + 6) + 8 = 2 + (6 + 8) 8 + 8 = 2 + 2 4 = 4 True If we were to try other elements, we would have the same result. It is associative under addition. 5. Commutative property: for all value of a and b, does a + b = b + a? 5 + 8 = 8 + 5 1 = 1 True 9 + 6 = 6 + 9 3 = 3 True If we were to try other elements, we would have the same result. The commutative property is true. This system satisfies the five properties required for a mathematical system to be a commutative group. Thus, clock 12 arithmetic under the operation of addition is a commutative, or abelian, group. A Few Things to Note If not every element in the set appears in every row and column of the table, however, you need to check the associative property carefully. If the elements are symmetric about the main diagonal, then the system is commutative. It is possible to have groups that are not commutative: noncommutative or nonabelian groups. Example : Investigating a System of Symbols Use the mathematical system defined by the table (below) and determine a) the set of elements. b) the binary operation. c) closure or nonclosure of the system. d) the identity element. e) the inverse of @. f) (@ # P) # P and @ # (P # P). g) W # P and P # W. .
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