1. Sets and Set Operations

2. Set – A collection of objects
example:
a set of tires
Element – An object contained within a set
example:
my car’s left front tire

3. 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, …}

4. 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

5. 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

6. 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

7. Empty Set – A set that
contains no elements
Notation: { } or
Universal Set – A set that
contains all of the elements
being considered
Notation: U

8. Complement of a set – A set that contains all of
the elements of the universal set that are not in
a given set
Notation: B means the complement of B

9. 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 {1, 2, 3, 4, 5}
Which sets are finite? n(E) = 4
A, C, D, E, F, G n(G) = 5
Which sets are equal to set C? F, G
Which sets are equivalent to set A? E

10. U = {1, 2, 3, 4, 5, 6, 7}
M = {2, 4, 6}
What is M ?
{1, 3, 5, 7}

11. Is { } the same as ? Yes
Is { } the same as ? No

12. 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 W True
Y W False
Z W True The empty set is a
W True subset of every set

13. 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 W True
Y W False
Z W False

14. How many subsets can a set have?
Number of Number of
Set Elements Subsets Subsets
{a} 1 {a},{ } 2
{a, b} 2 {a},{b},{a,b},{ } 4
{a, b, c} 3 {a},{b},{c},{a,b},
{a,c},{b,c},{a,b,c}, 8
{ }
{a, b, c, d} 4 16
n 2n
If a set has n elements, it has 2n subsets

15. How many proper subsets can a set have?
Number of
Number of Proper
Set Elements
Proper Subsets
Subsets
{a} 1 X
{a},{ } 1
{a, b} 2 X
{a},{b},{a,b},{ } 3
{a, b, c} 3 {a},{b},{c},{a,b},
X
{a,c},{b,c},{a,b,c}, 7
{ }
{a, b, c, d} 4 15
n 2n – 1
If a set has n elements, it has 2n – 1 proper
subsets

16. 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

18. A Venn Diagram allows us to organize the
elements of a set according to their
attributes.

20. U = {1, 2, 3, 4, 5, 6.5}
even 1 odd
4
3
2 5
6.5
prime

21. small blue
triangle
National Library of Virtual Manipulatives Attribute Blocks

22. Set Operations
The intersection of sets A and B is the set of all
elements in both sets A and B
notation: A B

23. 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

24. 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

25. 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

27. A = {1, 2, 3, 4, 5} B = {2, 4, 6} C = {3, 5, 7}
A B = {2, 4} A – B = {1, 3, 5}
A B = {1, 2, 3, 4, 5, 6} C – A = {7}
C B = {2, 3, 4, 5, 6, 7)
C B= { }
The set complement X – Y is the set of all
elements of X that are not in Y

28. Representing sets with Venn diagrams
A B A B
1 2 3
1 2 3
5
4 6
4
7
8
Two attributes C
22 or 4 regions
Three attributes
23 or 8 regions

29. A B A B
A A

30. A B A B
AUB A B
A B
A B

31. A B A B
AUB A B

32. A B
C
(A U B) C

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

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

35. A B
1 3
2
4
How many stars are in:
Circle A 3 Either A or B 7
Circle B 5 Exactly one circle 6
Only Circle A 2 Neither circle 2
Both A and B 1 Total stars = 9

36. 20
Out of 20 students:
B F 8 play baseball
7 play football
5 3 4 3 play both sports
8
How many play neither sport? 8
How many play only baseball? 5
How many play exactly one sport? 5 + 4 = 9

37. 30 Out of 30 people surveyed:
20 like Blue
B P 20 like Pink
4 4 5 15 like Green
10 14 like Blue and Pink
2 1
11 like Pink and Green
2 12 like Blue and Green
2 G 10 like all 3 colors
How many people like only Pink? 5
How many like Blue and Green but not Pink? 2
How many like none of the 3 colors? 2
How many like exactly two of the colors? 4 + 2 + 1 =7