MDFP Introduction to Mathematics SETS and Venn Diagrams

The following examples should help you understand the notation, terminology, and concepts related to Venn diagrams and set notation.

1. Words: N is the set of natural numbers or counting numbers. 2. List: N = {1, 2, 3, …} 3. Set-builder notation: N = {x | x  N}

A finite set has a limited number of members. Example: The set of students in our Math class. An infinite set has an unlimited number of members. Example: The set of integers. A well-defined set has a universe of objects which are allowed into consideration and any object in the universe is either an element of the set or it is not.

One way to represent or visualize sets is to use Venn diagrams:

Let U be the set of all students enrolled in classes this semester. U

 Let M be the set of all students enrolled in Math this semester.  Let E be the set of all students enrolled in English this semester. U M E

E  M = the set of students in Math AND English U E M

E  M = the set of students in Math OR English U EM

Let C be the set of all students enrolled in classes this semester, but who are not enrolled in Math or English, C = M U E U M C E

Two sets with no elements in common are called disjoint sets. U Students who enjoy Math Students who loathe Math

X is a subset of Y if and only if every member of X is also a member of Y. U Students in a Math class Students who enjoy Math Y X

Elements A = { 1, 2, 3, 4, 5, 6, 7, 8} B = {2, 4, 6, 8, 10} 2, 4, 6, 8 belong in BOTH A and B. A B = 2,4,6,8 

Elements A = {1, 2, 3, 4, 5, 6, 7, 8, 9} B = {2, 4, 6, 8} Therefore B is a subset of A ALL elements in B belong in A B A 

Elements A = {2, 4, 6, 8, 10} B = {1, 3, 5, 7, 9} Sets A and B are DISJOINT

U = {1, 2, 3, 4, 5, 6, 7, 8} A = {1, 2, 6, 7} B = {2, 3, 4, 7} C = {4, 5, 6, 7} A = {1,2,6,7}

U = {1, 2, 3, 4, 5, 6, 7, 8} A = {1, 2, 6, 7} B = {2, 3, 4, 7} C = {4, 5, 6, 7} B = {2, 3, 4, 7}

U = {1, 2, 3, 4, 5, 6, 7, 8} A = {1, 2, 6, 7} B = {2, 3, 4, 7} C = {4, 5, 6, 7}

Given the following sets: U = {1, 2, …., 16}, A = {3, 6, 9, 12, 15} and B = {factors of 12 } Find the following: a) A B b) EXAMPLE 1

A B 16 B = {factors of 12} = {1, 2, 3, 4, 6, 12}A = {3, 6, 9, 12, 15} and Draw Venn diagram Which numbers belong in A AND B ? A ∩ B Which numbers belong in NEITHER A OR B ? COMPLETE DIAGRAM !

A B C Example 2 Given the Venn diagram below find the following n(B  C) a) n( ) b) n(AB C ) c) NOTE: First complete the diagram!! The total elements should be 100, so

A B C Solution n(B  C) a) Tick every thing in set B We are looking for the number of elements in BOTH B and C, so Of those already ticked, which are in C? = = 7 (ie there are 7 elements) 37

A B C n ( ) b) 37 Find the number of elements NOT in B Tick all the elements in B Total the none ticked numbers = = 66

A B C c) n(A ) 37 Start at the left: Tick all elements in A A B means A OR B, so now tick all elements in B that have not already been ticked Now tick all elements not already ticked that DO NOT belong in C = 100 – 15 = 85

Complete All Questions Sets & Venn Diagrams 1 Any work not completed during class must be completed for homework !