Chipola College
MGF 1107
10.2 Finite Mathematical Systems
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A finite mathematical system is one whose set contains a finite number of elements.
Example 1/
Let’s develop a finite mathematical system called clock arithmetic.
What will be the finite set of elements used?
What will be the binary operation used?
Fill in the chart for Clock Arithmetic:
+
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
Is the clock arithmetic system a commutative group (abelian group)? Remember you must
show that all five conditions hold true.
1. Closure
2. Identity Element
3. Inverse Element
4. Associative Property (Normally you will not be asked to check every case. If every
element in the set does not appear in every row and column of the table, however;
you need to check the associative property carefully.
5. Commutative Property (If the elements are symmetric about the main diagonal, then
the system is commutative. The main diagonal is the diagonal from the upper lefthand corner to the lower right-hand corner of the table.)
Example 2/
Consider the following mathematical system.
given operation.
O 1 3
1 5 7
3 7 1
5 1 3
7 3 5
Assume the associative property holds for the
5
1
3
5
7
7
3
5
7
1
a) List the elements in the set of this mathematical system.
b) Identify the binary operation.
c) Determine whether it is a commutative group.
Example 3/
Use the mathematical system defined by the following and determine
@
@ P
P W
W @
P
W
@
P
W
@
P
W
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) W
P and P W
g) (@ P)
P and @
(P
P)