1. Section 10.2 Finite Mathematical Systems

2. What You Will Learn Finite Mathematical Systems: Clock Arithmetic
Mathematical Systems without Numbers

3. Definition
A finite mathematical system is one whose set contains a finite number of elements.

4. 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, = 1.
Also, = 1.
An addition table is found on the next slide.

5. Clock 12 Arithmetic

6. 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}

7. 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

8. Example 1: A Commutative Group?
Solution
3. Inverse elements: The additive inverse is the number that when added will yield the additive identity, is the additive identity of 4; = 12. And 4 is the additive identity of 8; = 12. Similarly, 7 and 5 are additive identities of each other. Here is the table of additive identities:

9. Example 1: A Commutative Group?
Solution

10. 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.

11. 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.

12. 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.

13. 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.

14. Example 3: Investigating a System of Symbols
Use the mathematical system defined by the table (next slide) 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
f) # P) # P # (P # P).
g) W # P and P # W.

15. Example 3: Investigating a System of Symbols
Solution
a) The set of elements is P, W}
b) The binary operation is #.
c) The table contains only the P, and W, so it is closed.

16. Example 3: Investigating a System of Symbols
Solution
d) The identity element is W.
e) The inverse is the element that when # yields the identity element, W. P is the inverse
@ # P = P = W

17. Example 3: Investigating a System of Symbols
Solution
f) # P) # P = W # P = P
and
@ # (P # P) = P

18. Example 3: Investigating a System of Symbols
Solution
g) W # P = P and P # W = P