A Mathematical System Consists of a Set of Elements and at Least One Binary Operation

ChipolaCollege

MGF 1107

10.1 Groups ______

A mathematical system consists of a set of elements and at least one binary operation.

An infinite mathematical system consists of a neverending number of elements in a set.

For example, the set of integers is infinite.

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. Examples/ addition, subtraction,

multiplication, and division

A set is closed under the given binary operation if a binary operation is performed on any

two elements of a set and the result is an element of the set.

1. Is the set of integers closed under addition?

1. Is the set of integers closed under multiplication?

1. Is the set of integers closed under division?

1. Is the set of integers closed under subtraction?

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.

1. What is the identity element for the set of integers under the operation of multiplication?(Multiplicative Identity)

1. What is the identity element for the set of integers under the operation of addition?(Additive Identity)

An inverseis when a binary operation is performed on two elements in a set and the result is

the identity element for the binary operation.

1. Does every integer have an inverse under the operation of multiplication?
2. Does every integer have an inverse under the operation of addition?

Properties of a Group:

Any mathematical system that meets the following four requirements is called a group.

1. The set of elements is closed under the given operation.
2. An identity element exists for the set under the given operation.
3. Every element in the set has an inverse under the given operation.
4. The set of elements is associative under the given operation.

Let’s consider the set of integers under the operation of addition. Is it a group? If not, which condition(s) fails?

Let’s consider the set of integers under the operation of multiplication. Is it a group? If not, which condition(s) fails?

A commutative group (or abelion group) is a group that satisfies the commutative property.

Let’s consider the set of integers under the operation of addition. Is it an abelian group?

Example 1 Is it a Group?

Determine whether the set of rational numbers under the operation of multiplication forms a group.

Practice:

1. Is the set of positive integers a commutative group under the operation of addition?

1. Is the set of integers a group under the operation of addition?

1. Is the set of negative integers a group under the operation of division?

1. Is the set of integers a group under the operation of multiplication?

1. Is the set of negative integers a group under the operation of multiplication?

1. Is the set of rational numbers a group under the operation of subtraction?

1. Is the set of positive integers a commutative group under the operation of division?