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Set Language and Notation

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Jonathan Foo

on 23 September 2012

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Transcript of Set Language and Notation

Set Language
and
Notation Set Language Notation Universal Set A set that contains all elements considered in a given discussion Denoted by Representation on Venn Diagram Elements Each object in a set is called the element of the set E.g. Set A, where 1 and 6 are elements of A Sets A set is a collection or group of things E.g. Natural Odd Numbers less than 10 we represent the above set with A set is denoted by a CAPITAL Letter O O = {1,3,5,7,9} How can we describe a set ? 1 2 3 Listing the elements Describing the elements Use set- builder Notation A = {2,3,5,7,11} C = {a,p,l,e}
*The order in which the elements are listed is NOT important. We can also write C = {a,e,l.p} B = {1,2,3,4,5, ...} A is the set of prime numbers less than 12 B is the set of positive integers C is the set of letters in the word 'apple' * Each element is listed only once A = {x:x is a prime number < 12} B = {x:x is a positive integer} C = {x:x is a letter of the word 'apple' } ??? If x is an element of set A, we write Is an element of x A If x is not an element of set A, we write Is NOT an element of x A Example A = {1, 3, 5, 7} 1 A 2 A Null Set Also known as an empty set It is denoted by { } or Example A = {x:x is a natural number 1 < Therefore A = Natural numbers are 1,2,3,4,5 ... *Important Note {0} is not an empty set since it contains one element Number of Elements The number of elements
in set A is denoted by n(A) Example 1 Example 3 Example 4 Example 2 A = {2, 4, 6, 8} n(A) = 4 B = {x : x is an odd number} B = {1, 3, 5, 7, 9 ...} There are 4 elements in this set n(B) = infinite * It is impossible to list all the elements in set B C = {x : x is a letter of the word 'happy'} c = {h, a, p, y} * Remember not to repeat identical elements n(C) = 4 D = {x:x is an integer and 2x=1} D = n(D) = 0 2x=1
x=0.5 Since x is not an integer, Therefore D is an empty set * Equal Sets Two sets are equal if they contain exactly the same elements Subset If every element of set A is also an element of set B, then set A is a subset of set B Proper SubSet If two sets A and B are equal, we write A B If two sets A and B are not equal, we write A B Example C = {x:x is a letter in the word 'apple'} D = {x:x is a letter in the word 'lead'} C= { a, p, l, e} D = [ l, e, a, d} C = D It is written as Example If set A is not a subset of B, it is written as A = {1, 2, 3}
B = {1, 2, 3, 4} since every element in A is also a member in B A B However, B A since not every element of B is in A * if A B and B A , then A and B have exactly the same elements A B Set A is a proper subset of B, If 1) Every element of set A is also an element of set B 2) Set B has more elements than set A Example A = {1, 2, 3}
B = {1, 2, 3, 4} A B Important Tip * The empty set/Null set is the subset for every set * Any subset is a subset of it self Example: List the subsets of set A, where A = {1, 5} {1} , {5} , { } and {1,5} The subsets are Venn Diagram Complement of a set Written A' is the set of all elements that are in the Universal set that are not in A Example = {1, 2, 3 ,4} A = {1, 2, 3} A' = {4} A large Rectangle is used to represent the Universal Set Circles or ovals are drawn inside the rectangle to represent the subsets of Example Examples of Venn Diagrams Disjoint Sets Two sets which have no elements in common 2 1 4 3 Union of Sets The union of 2 sets A and B is the set of elements which are in A or in B or Both A and B Denoted by Intersection of of Sets The intersection of 2 sets, A and B is the set of elements which are common to both A and B Denoted by Elements of Sets in Venn Diagram AceLearning Under topic on
Set Language and Notation Drawing of Venn Diagram AceLearning Under topic on
Set Language and Notation Maximum & Minimum Values n(A)=24 n(B)=17 n( ) =40 n( ) n( ) Maximum Value Minimum Value Minimum Value Maximum Value A B A B A A B 24-x x 17-x n(B) = 17 n(A) = 24
24-x+x+17-x = 40
x =1 7 17 16 There must be 17 in B and 24 in A. Hence to get Maximum possible value for intersection fof A and B. B must be a subset of A . In this case the intersection is only B n(B) = 17 n(A) = 24
Since B is a subset of A
n( ) = 1 n( ) = 17 17 7 16 7+17= 24 n( ) = 24 The union of A and B in this CAse is only A. B 23 1 16 In this case the intersection of Set A and B is the total amount present n( ) = n( ) =40 ARe you
all
Set? Set Notation & Venn Diagram Video Tutorial n(B)= 17 n(A) = 24 n(B) = 17 n(A) = 24
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