1150 day 2
TRANSCRIPT
Sets and Set Operations
Set – A collection of objectsexample: a set of tires
Element – An object contained within a setexample: my car’s left front tire
Finite set – Contains a countable number of objects
Example: The car has 4 tires
Infinite set- Contains an unlimited number of objects
Example: The counting numbers {1, 2, 3, …}
Cardinal Number – Used to count the objects in a set
Example: There are 26 letters in the alphabet
Ordinal Number – Used to describe the position of an element in a set
Example: The letter D is the 4th letter of the alphabet
Equal sets – Sets that contain exactly the same elements (in any order)
{A, R, T, S} = {S, T, A, R}
Notation: A = B means set A equals set B
Equivalent sets – Sets that contain the same number of elements (elements do not have to be the same)
{C, A, T} ~ {d, o, g}
Notation: A ~ B means set A is equivalent to set B
Empty Set – A set that contains no elements Notation: { } or
Universal Set – A set that contains all of the elements being considered
Notation: U
Complement of a set – A set that contains all of the elements of the universal set that are not in a given set
Notation: means the complement of BB
A = {2, 4, 6, 8} B = {1, 2, 3, 4, …}C = {1, 2, 3, 4, 5} D = { }E = {Al, Ben, Carl, Doug} F = { 5, 4, 3, 2, 1}G = {x | x < 6 and x is a counting number}
Set Builder Notation
Which sets are finite?
Which sets are equal to set C?
Which sets are equivalent to set A?
{1, 2, 3, 4, 5}
n(E) =n(G) =
45
F, G
E
A, C, D, E, F, G
U = {1, 2, 3, 4, 5, 6, 7}
M = {2, 4, 6}
What is ?M
{1, 3, 5, 7}
Is { } the same as ? Yes
Is { } the same as ? No
Set B is a subset of set A if every element of set B is also an element of set A.
Notation: B A
W = {1, 2, 3, 4, 5} X = {1, 3, 5}Y = {2, 4, 6} Z = {4, 2, 1, 5, 3}
True or False:X WY WZ W W
TrueFalseTrueTrue
The empty set is a subset of every set
Set B is a proper subset of set A if every element of set B is also an element of set A AND B is not equal to A.
Notation: B A
W = {1, 2, 3, 4, 5} X = {1, 3, 5}Y = {2, 4, 6} Z = {4, 2, 1, 5, 3}
True or False:X WY WZ W
TrueFalseFalse
How many subsets can a set have?
Set{a}
{a, b}{a, b, c}
{a, b, c, d}
Number of Elements
123
4n
SubsetsNumber of
Subsets24
8
162n
{a},{ }{a},{b},{a,b},{ }{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c},{ }
If a set has n elements, it has 2n subsets
How many proper subsets can a set have?
Set{a}
{a, b}{a, b, c}
{a, b, c, d}
Number of Elements
123
4n
Proper Subsets
Number of Proper Subsets
13
7
152n – 1
{a},{ }{a},{b},{a,b},{ }{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c},{ }
If a set has n elements, it has 2n – 1 proper subsets
XX
X
W = {a, b, c, d, e, f}
How many subsets does set W have?26 = 64
How many proper subsets does set W have?26 – 1 = 64 – 1 = 63
A Venn Diagram allows us to organize the elements of a set according to their attributes.
U = {1, 2, 3, 4, 5, 6.5}
even odd
prime
1
23
4
5
6.5
National Library of Virtual Manipulatives Attribute Blocks
small blue
triangle
Set Operations
The intersection of sets A and B is the set of all elements in both sets A and B
notation: A B
The union of sets A and B is the set of all elements in either one or both of sets A and B
notation: A B
The union of sets A and B is the set of all elements in either one or both of sets A and B
notation: A B
The union of sets A and B is the set of all elements in either one or both of sets A and B
notation: A B
A = {1, 2, 3, 4, 5} B = {2, 4, 6} C = {3, 5, 7}
A B =
A B =
C B =
C B =
{2, 4}
{1, 2, 3, 4, 5, 6}
{2, 3, 4, 5, 6, 7)
{ }
The set complement X – Y is the set of all elements of X that are not in Y
A – B =
C – A =
{1, 3, 5}
{7}
Representing sets with Venn diagrams
A B A B
C
Three attributes23 or 8 regions
1 2 3
4
1 2 3
45
6
78
Two attributes22 or 4 regions
A B
A
A B
A
A B
A U B
A B
A BA B
A B
A B
A U B
A B
A B
A B
C
(A U B) C
(A U B) C
A B
C
1 2 3
45
6
78
A = B = C = C =
A U B =
(A U B) C =
{1, 2, 4, 5}{2, 3, 5, 6}{4, 5, 6, 7}{1, 2, 3, 8}
{1, 2, 3, 4, 5, 6}
{1, 2, 3}
A U (B C)
A B
C
A = B = C = B C =
A U (B C) =
{3, 6, 7, 8}{2, 3, 5, 6}{4, 5, 6, 7}
{5, 6}
{3, 5, 6, 7, 8}
3
78
65
4
1 2
A B1 2
3
4
How many stars are in:Circle ACircle BOnly Circle ABoth A and B
Either A or BExactly one circleNeither circleTotal stars =
3521
7629
B FOut of 20 students:
8 play baseball7 play football3 play both sports
How many play neither sport?
How many play only baseball?
How many play exactly one sport?
20
35 4
8
8
5
5 + 4 = 9
B P
G
Out of 30 people surveyed:20 like Blue20 like Pink15 like Green14 like Blue and Pink11 like Pink and Green12 like Blue and Green10 like all 3 colors
How many people like only Pink?How many like Blue and Green but not Pink?How many like none of the 3 colors?How many like exactly two of the colors?
102 1
4
2
54
2
30
52
24 + 2 + 1 =7