Download Venn Diagrams Notation File

MDFP
Introduction to Mathematics
SETS and Venn
Diagrams
SETS AND VENN
DIAGRAMS
 A set is a collection of objects.
 There are three ways to describe
a set:
VENN DIAGRAMS & SET NOTATION
The following examples should help
you understand the notation,
terminology, and concepts related
to Venn diagrams and set notation.
EXAMPLES OF SETS
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}
KINDS OF SETS
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.
VENN DIAGRAMS
One way to represent or visualize sets is to use
Venn diagrams:
UNIVERSAL SET
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
INTERSECTION []
E  M = the set of students in Math AND English
U
E
M
INTERSECTION []
UNION []
E  M = the set of students in Math OR English
U
E
M
COMPLEMENT OF A SET
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
C
M
E
DISJOINT SETS
Two sets with no elements in common are called
disjoint sets.
U
Students who
enjoy Math
Students who
loathe Math
SUBSET ()
X is a subset of Y if and only if every member of X
is also a member of Y.
U
Y
Students in a Math class
X
Students who
enjoy Math
VENN DIAGRAM EXAMPLE 1
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
VENN DIAGRAM EXAMPLE 2
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
VENN DIAGRAM EXAMPLE 3
Elements
A = {2, 4, 6, 8, 10}
B = {1, 3, 5, 7, 9}
Sets A and B are DISJOINT
VENN DIAGRAM EXAMPLE 4
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}
VENN DIAGRAM EXAMPLE 5
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}
VENN DIAGRAM EXAMPLE 6
U = {1, 2, 3, 4, 5, 6, 7, 8}
A = {1, 2, 6, 7}
B = {2, 3, 4, 7}
C = {4, 5, 6, 7}
C = {4, 5, 6, 7}
EXAMPLE 1
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)
AB
• Draw Venn diagram
5
7
A
8
10
9
15
11
3
6
12
13 14 16
B
4
1
2
16
• Which numbers belong in A AND B ? A
∩B
• Which numbers belong in NEITHER A OR B?
AB
COMPLETE DIAGRAM !
A = {3, 6, 9, 12, 15} and B = {factors of 12} = {1, 2, 3, 4, 6, 12}
Example 2
Given the Venn diagram below find the following
a) C) n(B
b)
B)
c) n(A  B
 C)
n(
NOTE: First complete the diagram!! The total elements should be 100, so ......
100
A
10
20
7
2
4
5
B
15
37
C
Solution
a) C)n(B = 2 + 5 = 7 (ie there are 7 elements)
• 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?
B
A
7
10

2
4
37

15
C
20 
5

100
n B ) = 10 + 4 + 15 + 37 = 66
b)
(
Find the number of elements NOT in B
Tick all the elements in B
Total the none ticked numbers
B
A
7
10

2
4
15
37
C
20 
5

100
c) n(A  B C) = 100 – 15 = 85
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
B
A
10
7

4
37 

2
15
C
20
5


100
Complete
All Questions
Sets & Venn Diagrams 1
Any work not completed during class must
be completed for homework !