Finite Mathematical Systems Section 9.2 Finite Mathematical Systems

What You Will Learn Upon completion of this section, you will be able to: Determine whether a finite mathematical system defined by clock arithmetic is a group. Determine whether a finite mathematical system without numbers is a group.

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 found on the next slide.

Clock 12 Arithmetic

Example 1: A Commutative Group? 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}

Example 1: A Commutative Group? 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

Example 1: A Commutative Group? Solution 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:

Example 1: A Commutative Group? Solution

Example 1: A Commutative Group? Solution 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.

Example 1: A Commutative Group? Solution 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.

Example 1: A Commutative Group? Solution 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 6: Is the System a Commutative Group? Determine whether the mathematical system defined by the table is a commutative group under the operation of .

Example 6: Is the System a Commutative Group? Solution The system is closed. There is an identity element, A. Each elements has an inverse; A is its own inverse, and B and C are inverses of each other.

Example 6: Is the System a Commutative Group? Solution Every element in the set does not appear in every row and every column of the table, so we need to check the associative property carefully. The following counterexample illustrates that the associative property does not hold for every case.

Example 6: Is the System a Commutative Group? Solution The commutative property holds because there is symmetry about the main diagonal. Since we have shown that the associative property does not hold under the operation of , this system is not a group. Therefore, it cannot be a commutative group.