1. Unit 1
Sets
§2.1: Symbols and Terminology ..........................................................................................3
§2.2: Venn Diagrams and Subsets .......................................................................................7
§2.3: Set Operations ...........................................................................................................13
§2.4: Surveys and Cardinal Numbers ................................................................................21

3. §2.1: SYMBOLS AND TERMINOLOGY
Definition:
A set is a ______________ of objects. If an object belongs to the set, we call it an
______________ or ______________.
Notation:
x ∈ A means x is ____ ___________ of A.
x ∉ A means x is ____ ____ ___________ of A.
Remark
A set must be ______________________. This means that we can determine if an
element belongs to the collection.
Naming Sets
There are three ways to name a set:
•
Listing (Roster Form)
The elements of the set are listed, separated by ______________. The entire list is
enclosed in braces,
•
{ }.
The order of the elements is not important.
Word Description (Verbal Description)
The description provided must be ______________.
•
Set-builder Notation
Set-builder notation encloses a general ______________ for the elements and a
description of any restrictions on the formula input.
{ formula for element | restriction on formula's input}
Example 1:
Write a verbal description of the set.
A = {2,3,5, 7,11,13,17,19}
Solution:
Unit 1: Sets
3

4. Example 2:
List the elements in the set.
B = { x 2 − 1| x ∈ N and x ≤ 5 }
Solution:
Example 3:
Identify each collection as “well defined” or “not well defined”.
A.
{ x | x is a book catalogued by JALC}
B.
{ x | x is a good movie}
C.
{ x | x is a counting number greater than 4}
Solution:
A.
B.
C.
4
Unit 1: Sets

5. Example 4:
List the elements of the set.
A is the set of odd counting numbers greater than 10.
Solution:
Example 5:
List the elements of the set.
A is the set of all astronauts who have walked on Neptune.
Solution:
Two Special Sets
Definition:
The ____________ ______ is the set that contains no elements.
Notation:
We can write _____ or _______.
Definition:
The ________________ ________ is the set that contains all of the elements
relevant to the context.
Notation:
We write _______.
Unit 1: Sets
5

6. Equal Sets
Definition:
Two sets are said to be equal if they have the __________ ________________.
Notation:
If A and B are equal, then we write A = B.
Cardinality of a Set
Definition:
The ______________ of _________________ in a set is called the cardinal
number of a set.
Notation:
n ( A ) = cardinal number of A
Example 6:
Find n ( A ) .
A = { x | x is a vowel in the English alphabet}
Solution:
Example 7:
Find n ( B ) .
B = { x | x is two-digit natural number}
Solution:
6
Unit 1: Sets

7. §2.2: VENN DIAGRAMS AND SUBSETS
Venn Diagrams
Definition:
A Venn diagram is a ___________ _______________ of a set or sets.
Venn diagrams will be particularly useful when we discuss set operations in §2.3.
Example 1:
In the Venn diagram below, each region is given a number.
U
B
A
2
1
3
4
A. List the numbered regions which correspond to the set A.
B. List the numbered regions which correspond to the set B.
C. List the numbered regions which correspond to the set U.
Solution:
A.
B.
C.
Unit 1: Sets
7

8. Example 2:
Shade the regions in the Venn diagram which correspond to the elements
that are not in A.
U
B
A
2
1
3
4
Complement of a Set
Definition:
The complement of a set A is the set of all elements that are ________ _____ the
set A.
Notation:
A′ = the complement of A.
Remark:
The regions we shaded in Example 2 correspond to A′.
Example 3:
Write the definition of the complement of the set A in set-builder notation.
Solution:
8
Unit 1: Sets

9. Example 4:
Given U = {rain, snow, sleet, hail, wind, frost, fog} and A = {snow, wind, frost} ,
find A′.
Solution:
Example 5:
Find each of the following.
A. U ′
B. ∅′
C.
( A′)′
Solution:
A.
B.
C.
Unit 1: Sets
9

10. Subsets
Example 6:
When you order a hamburger at PawPaw J’s Hamburger Grill, you can pick from
the following set of condiments.
{ketchup, mustard, pickles}
Write out sets that correspond to every possible choice of condiments.
Solution:
Definition:
We say the set A is a subset of the set B if ____________ _________________
that belongs to A ___________ _______________ to B.
Notation:
A ⊆ B means A is a subset of B.
Example 7:
List all of the subsets of the set {hot, cold, warm, frigid} .
Solution:
10
Unit 1: Sets

11. Formula:
Let A be a set with n elements. Then A has 2n subsets.
Note: Where did the two come from? Hint: Remember that a set must be well-defined.
Example 8:
Given A = {gold, silver, bronze, magnesium, copper} , how many subsets does the
set A have?
Solution:
Two Unexpected Subsets
Example 9:
Let A be a nonempty set. What are two sets that are guaranteed to be subsets of A?
Solution:
Definition:
If A is a subset of B and A ≠ B, we call A a _____________ _______________
of B.
Notation:
A ⊂ B means A is a proper subset of B.
Unit 1: Sets
11

12. Formula:
Let A be a set with n elements. Then A has __________ proper subsets.
Example 10: Let T = {Edwards, Owen, Calvin, Luther, Hodge, Augustine} . How many proper
subsets of T are there?
Solution:
12
Unit 1: Sets

13. §2.3: SET OPERATIONS
Example 1:
In the Venn diagram below, shade the region(s) that correspond to the elements
that are in both A and B.
U
B
A
2
1
3
4
Example 2:
Neldys will be happy with any combination of the following set of toppings for a
pizza
N = {pepperoni, sausage, mushrooms, olives, peppers} ,
and Jim will be happy with any combination of the following set of toppings for a
pizza
J = {sausage, onions, tomatoes, olives, bacon} .
With what set of toppings would they both be happy?
Solution:
Unit 1: Sets
13

14. Intersections
Definition:
The intersection of the set A and the set B is the set of those elements that are in
both A and B.
Notation:
A ∩ B means the intersection of A and B.
Remark:
You will notice that we shaded the region in Example 1 that corresponds to
A ∩ B.
Example 3:
Write the definition of A ∩ B in set-builder notation.
Solution:
Example 4:
Let U = {Freud, Jung, Piaget, Skinner, Bandura, Rogers, Pavlov, Lewin, Erikson, James} ,
A = {Freud, Piaget, Skinner, Jung} , and B = {Piaget, Jung, Rogers, Lewin} . Find A′ ∩ B.
Solution:
14
Unit 1: Sets

15. Unions
Example 5:
Debbie rolls a single, six-sided die. She will win a necklace if she rolls an even
number or a number less than 3. Write the list of outcomes for which she would
win.
Solution:
Definition:
The union of the set A and the set B is the set of those elements that are in A or B
(or both).
Notation:
A ∪ B means the union of A and B.
Example 6:
Write the definition of A ∪ B in set-builder notation.
Solution:
Example 7:
In the Venn diagram below, shade the region(s) that correspond to A ∪ B.
U
B
A
2
1
3
4
Unit 1: Sets
15

16. Example 8:
Let U = {a, b, c, d , e, f , g , h, i, j} , A = {a, b, d , g , h} , and B = { g , h, i, j} . Find each
of the following.
A. A′
B. B′
C. A ∪ B
D. A ∩ B
E.
F.
G. A′ ∪ B′
H. A′ ∩ B′
( A ∪ B )′
A.
B.
C.
D.
E.
F.
G.
H.
16
( A ∩ B )′
Unit 1: Sets

17. Rule: DeMorgan’s Law for Sets
•
( A ∪ B )′ =A′ ∩ B′
•
( A ∩ B )′ =A′ ∪ B′
Differences
Definition:
The difference of the sets A and B is the set of those elements that are in A but not
B.
Notation:
A − B means the difference of A and B.
Example 9:
Write the definition of A − B in set-builder notation.
Solution:
Example 10: In the Venn diagram below, shade the region(s) that correspond to A − B.
U
B
A
2
1
3
4
Unit 1: Sets
17

18. Example 11: Let U = {a, b, c, d , e, f , g , h, i, j} , A = {a, b, d , g , h} , and B = { g , h, i, j} . Find each
of the following.
A. A − B
B. B − A
C. U − A
D. What is another name for U − A ?
A.
B.
C.
D.
Warning:
18
A − B does not mean the same thing as B − A.
Unit 1: Sets

19. Shading Venn Diagrams
Use the numbered regions to help you determine which regions to shade. Think of it as “paintby-number.”
Example 12: In the Venn diagram below, shade the region(s) corresponding to A′ ∩ B.
U
B
A
2
1
3
4
Solution:
Unit 1: Sets
19

20. Example 13: In the Venn diagram below, shade the region(s) corresponding to ( A ∪ B′ ) ∩ C.
U
B
A
C
Solution:
20
Unit 1: Sets

21. §2.4: SURVEYS AND CARDINAL NUMBERS
Cardinality when Two Sets are Involved
Example 1:
In a group of 47 students, 22 students are enrolled in a science class, 28 are
enrolled in a humanities class, and 7 are enrolled in both.
A. How many students are enrolled in a science class or a humanities class?
B. How many are enrolled in neither?
Solution:
Example 2:
Find the value of n ( A ∪ B ) = 19, n ( B ) 14, and n ( A ∩ B ) =
if n ( A ) =
4.
Solution:
Unit 1: Sets
21

22. Developing a Formula for n ( A ∪ B )
U
B
A
2
1
3
4
In the above Venn diagram,
•
Determine the regions accounted for in each of the following.
o
o
n ( B ) counts the elements in regions ____________
o
n ( A ) + n ( B ) counts the elements in regions ____________ and ________
o
•
n ( A ) counts the elements in regions ____________
n ( A ∪ B ) counts the elements in regions ____________
Does the formula n ( A ∪ B ) n ( A ) + n ( B ) make sense? Explain why or why not. ______
=
________________________________________________________________________
________________________________________________________________________
•
If we want to count the elements in region 2, what notation would we use? ___________
Formula:
For any two sets A and B,
n ( A ∪ B) =
_______________________________
22
Unit 1: Sets

23. Example 3:
Find the value of n ( A ∩ B ) = 30, n ( B ) 19, and n ( A ∪ B ) =
if n ( A ) =
43.
Solution:
Cardinality when Three Sets are Involved
Example 4:
Use the given Venn diagram and the given information to fill in the number of
elements in each region.
n ( A ∩ B ) 21, n ( A ∩ B ∩ C ) 6, n ( A ∩ C ) 26, n ( B ∩ C ) 7,
=
=
=
=
n ( A ∩ C ′ ) 20, n ( B ∩ C ′ ) 25, n (= 40, and n ( A′ ∩ B′ ∩ C ′ ) 2.
C)
=
=
=
U
B
A
C
Unit 1: Sets
23

24. Example 5:
A survey of 80 movie renters was taken.
25
45
32
10
12
7
4
enjoy horror films.
enjoy romantic comedies.
enjoy documentaries.
enjoy horror films and romantic comedies.
enjoy romantic comedies and documentaries.
enjoy horror films and documentaries.
enjoy all three.
Use a Venn diagram to answer each question.
A. How many enjoy documentaries and romantic comedies, but not horrors?
B. How many enjoy none of these of three types of movies?
24
Unit 1: Sets