The document discusses sets and set operations including defining sets, elements, cardinal and ordinal numbers, equal and equivalent sets, subsets, Venn diagrams, and set operations like union, intersection, and complement. Examples are provided to illustrate concepts like finite and infinite sets, subsets, Venn diagrams representing multiple sets and operations, and using Venn diagrams to solve problems involving sets and their relationships.

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