## Mathematical Systems

A mathematical system consists of a set of elements and one or more binary operations to connect these elements.

### Unary Operation

A unary operation is an operation on a single element.

e.g. Squaring 4

finding the square root of 9

### Symbols

A binary operation is an operation that combines two elements of a set to give a single element.

e.g. multiplication 3 x 4 = 12

matrix subtraction ### Properties of Systems

Suppose a system consists of a set S and the binary operation *. The system can have the following properties.

 Property Description Examples Closure Set S is closed if all of the elements of set S combine to give an answer also in set S a * b = c where a, b, c S Identity element An identity element leaves every element of the set S unchanged. a * i = a i * a = a where i is the identity element. Inverse element An element and its inverse combine to give the identity element. a * b = i b * a = i where a and b are inverses. Commutative property An operation is commutative if the order in which the elements are combined does not change the result. a * b = b * a Associative property An operation is associative if, when combining the elements, the grouping of the elements does not change the result. (a * b) * c = a * (b * c) Distributive property Given two operations * and •, then * is distributive over • as in the example. a * (b • c) = (a * b) • (a * c)

Examples

 Example 1 Answers Give examples to show the following properties for the system (I, +): a. Closure 5 + -2 = 3, 3 I b. Identity element 3 + 0 = 3, 0 is the identity element. c. Inverse 3 + -3 = 0, 3 and -3 are inverses. d. Commutativity 3 + -2 = -2 + 3 e. Associativity (3 + 4) + 5 = 3 + (4 + 5) Example 2 Answers Give examples of systems that are not: a. Commutative (I, -) e,g, 3 − 4 is not equal to 4 − 3 b. Closed (W, ÷) e.g. 3 ÷ 4 = 0.75, 0.75 W

### Groups

Groups are mathematical systems that possess certain properties.
A mathematical system (S, *) is a group if it possesses the following properties:

a. Set S is closed under the operation *.
b. There is an identity element in set S.
c. Every element has an inverse in set S.
d. The associative property applies to the system.

### Tables

Many systems can be shown in tables. For the system ({a, b, c}, *)

 2nd 1st * a b c a a b c b b c a c c a b

For a system to be a group, every element must occur only once in each row and column of the table. The group properties can easily be tested from these tables.

a. Closure Every element in the table is in the set {a, b, c}
b. Identity element Look for a column that is in the same order as the original set. i.e. is the identity element.
c. Inverses Look for the identity element in each row.

a * a = a,
b * c = a,
c * b = a
This shows that each element has an inverse.

d. Associative property It is difficult to show that every combination of elements is associative. Check an example:

 Example (a * b) * c = a * (b * c) therefore b * c = a * a therefore a = a

Conclusion
{a, b, c}, *) is a group. As the group also shows commutativity it is called an Abelian orcommutative group.

### Modular Arithmetic

This is a type of number system where numbers are represented by the remainder after division by the modulo number.
e.g. In modulo 6, the number 8 is given as 2 (the remainder when 8 is divided by 6)

In modulo 4, 2 + 3 = 5, which is shown as 1 (the remainder when 5 is divided by 4.)

Draw up a table for the system ({0, 1, 2, 3}, addition modulo 4) and test to see if the system forms a group. You can assume the system is associative.
 + 0 1 2 3 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2

a. System is closed.
b. Identity element is 0.
c. Every element has an inverse.
Inverse of 0 is 0.
Inverse of 1 is 3.
Inverse of 2 is 2.
Inverse of 3 is 1.
d. It is associative (assumed)

Therefore the system forms a group.