10.1 ­ Groups ­ filled in.notebook December 01, 2015 10.1 ­ Groups 1 10.1 ­ Groups ­ filled in.notebook December 01, 2015 A mathematical system consists of a set of elements and at least one binary operation. 2 10.1 ­ Groups ­ filled in.notebook December 01, 2015 A mathematical system consists of a set of elements and at least one binary operation. A binary operation is an operation, or rule, that can be performed on two and only two elements of a set with the result being a single element. 3 10.1 ­ Groups ­ filled in.notebook December 01, 2015 A mathematical system consists of a set of elements and at least one binary operation. A binary operation is an operation, or rule, that can be performed on two and only two elements of a set with the result being a single element. Examples of binary operations: Addition, Subtraction, Multiplication, Division Examples of non­binary operations: Finding the reciprocal, inverse, identity 4 10.1 ­ Groups ­ filled in.notebook December 01, 2015 Commutative Property Addition Multiplication Associative Property Addition Multiplication 5 10.1 ­ Groups ­ filled in.notebook December 01, 2015 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 we say it has closure, under the given binary operation. 6 10.1 ­ Groups ­ filled in.notebook December 01, 2015 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 we say it has closure, under the given binary operation. 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. Examples: 3 + 0 = 3 18 + 0 = 18 3 x 1 = 3 18 x 1 = 18 The additive identity is zero. The multiplicative identity is one. 7 10.1 ­ Groups ­ filled in.notebook December 01, 2015 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 we say it has closure, under the given binary operation. 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. Examples: 3 + 0 = 3 18 + 0 = 18 3 x 1 = 3 18 x 1 = 18 The additive identity is zero. The multiplicative identity is one. 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 the be the inverse of the other. Examples: 3 + ­3 = 0 18 + ­18 = 0 1 1 3 x /3 = 1 18 x /18 = 1 The additive inverse of a number is its opposite. The multiplicative inverse of its number is its reciprocal. 8 10.1 ­ Groups ­ filled in.notebook December 01, 2015 PROPERTIES OF 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. 9 10.1 ­ Groups ­ filled in.notebook December 01, 2015 Is the set of integers a group under the operation of addition? Run through the check list to find out. 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. 10 10.1 ­ Groups ­ filled in.notebook December 01, 2015 Is the set of rational numbers a group under the operation of multiplication? Run through the check list to find out. 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. 11 10.1 ­ Groups ­ filled in.notebook December 01, 2015 Is the set of natural numbers a group under the operation of addition? Run through the check list to find out. 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. 12 10.1 ­ Groups ­ filled in.notebook December 01, 2015 A group that satisfies the commutative property is called a commutative group, or an abelian group. 13 10.1 ­ Groups ­ filled in.notebook December 01, 2015 A group that satisfies the commutative property is called a commutative group, or an abelian group. PROPERTIES OF AN ABELIAN 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. 5) The set of elements is COMMUTATIVE under the given operation. 14.