Section 9.1 Groups

What You Will Learn Upon completion of this section, you will be able to: Understand and identify mathematical systems. Understand and use the commutative and associative properties. Understand and identify closure. Understand and identify identity elements and inverses. Show whether a mathematical system is a group or a commutative group.

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

Binary Operation 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. Addition, multiplication, subtraction and division are all binary operations. The set of integers and the binary operation of addition make a mathematical system.

Commutative Property For any elements a, b, and c Addition a + b = b + a Multiplication a • b = b • a

Associative Property For any elements a, b, and c Addition (a + b) + c = a + (b + c) Multiplication (a • b) • c = a • (b • c)

Closure If a binary operation is performed on any two elements of a set and the result is an element of the set, then that set is closed (or has closure) under the given binary operation.

Identity Element 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. Additive identity element is 0. Multiplicative identity element is 1.

Inverses When a binary operation is performed on two elements in a set and the result is the identity element for the binary operation, each element is said to be the inverse of the other.

Inverses The additive inverse of a nonzero integer, a, is –a. 0 is its own additive inverse. The multiplicative inverse of a is 1/a. However, it may not be an integer; so most integers do not have a multiplicative inverse in the set of integers.

Properties of a Group Any mathematical system that meets the following four requirements is called a group. The set of elements is closed under the given operation. An identity element exists for the set under the given operation. Every element in the set has an inverse under the given operation. The set of elements is associative under the given operation.

Commutative Group A group that satisfies the commutative property is called a commutative group (or abelian group).

Properties of a Commutative Group A mathematical system is a commutative group if all five conditions hold. The set of elements is closed under the given operation. An identity element exists for the set under the given operation. Every element in the set has an inverse under the given operation. The set of elements is associative under the given operation. The set of elements is commutative under the given operations.

Example 1: Whole Numbers Under Addition Determine whether the mathematical system consisting of the set of whole numbers under the operation of addition forms a group.

Example 1: Whole Numbers Under Addition Solution 1. Closure: Sum of any two whole numbers is a whole number. The set is closed under addition. 2. Identity element: 0 is the additive identity. a + 0 = a. The system contains an identity element.

Example 1: Whole Numbers Under Addition Solution 3. Inverse elements: The additive inverse is the opposite of the number. –1 is the additive inverse of 1, –2 is that of 2, and so on. The numbers, –1, –2, … are not in the set of whole numbers. The mathematical system is not a group.

Example 3: Real Numbers Under Addition Determine whether the set of real numbers under the operation of addition forms a commutative group.

Example 3: Real Numbers Under Addition Solution 1. Closure: Sum of any two real numbers is a real number. The set is closed under addition. 2. Identity element: 0 is the additive identity. a + 0 = a + 0 = a. The system contains an identity element.

Example 3: Real Numbers Under Addition Solution 3. Inverse elements: The additive inverse is the opposite of the number. –1 is the additive inverse of 1, and –a is that of a. a +(–a) = –a + a = 0. All real numbers have an additive inverse.

Example 3: Real Numbers Under Addition Solution 4. Associative property: for any real numbers a, b and c, (a + b) + c = a + (b + c). 5. Commutative property: for any real numbers a and b, a + b = b + a. The set of real numbers forms a commutative group.