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

§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: means is ____ ___________ of . means is ____ ____ ___________ of .

x A x Ax A x 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.

{ }2,3,5,7,11,13,17,19A =

Solution:

Unit 1: Sets 3

Example 2: List the elements in the set.

{ }2 1| and 5 B x x N x= − ∈ ≤

Solution:

Example 3: Identify each collection as “well defined” or “not well defined”.

A. { }| is a book catalogued by JALCx x

B. { }| is a good moviex x

C. { }| is a counting number greater than 4x x

Solution:

A.

B.

C.

4 Unit 1: Sets

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

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: ( ) cardinal number of n A A=

Example 6: Find ( ).n A

{ }| is a vowel in the English alphabetA x x=

Solution:

Example 7: Find ( ).n B

{ }| is two-digit natural numberB x x=

Solution:

6 Unit 1: Sets

§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.

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.

U A B

1 2

3

4

Unit 1: Sets 7

Example 2: Shade the regions in the Venn diagram which correspond to the elements that are not in A.

Complement of a Set

Definition: The complement of a set A is the set of all elements that are ________ _____ the

set A.

Notation: the complement of .A 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:

U A B

1 2

3

4

8 Unit 1: Sets

Example 4: Given { }rain, snow, sleet, hail, wind, frost, fogU = and { }snow, wind, frost ,A = find .A′

Solution:

Example 5: Find each of the following.

A. U ′

B. ′∅

C. ( )A ′′

Solution:

A.

B.

C.

Unit 1: Sets 9

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

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 { }gold, silver, bronze, magnesium, copper ,A = 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

Formula: Let A be a set with n elements. Then A has __________ proper subsets.

Example 10: Let { }Edwards, Owen, Calvin, Luther, Hodge, Augustine .T = How many proper subsets of T are there?

Solution:

12 Unit 1: Sets

§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.

Example 2: Neldys will be happy with any combination of the following set of toppings for a pizza

{ }pepperoni, sausage, mushrooms, olives, peppers ,N = and Jim will be happy with any combination of the following set of toppings for a pizza

{ }sausage, onions, tomatoes, olives, bacon .J = With what set of toppings would they both be happy? Solution:

U A B

1 2

3

4

Unit 1: Sets 13

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 { }Freud, Jung, Piaget, Skinner, Bandura, Rogers, Pavlov, Lewin, Erikson, James ,U =

{ }Freud, Piaget, Skinner, Jung ,A = and { }Piaget, Jung, Rogers, Lewin .B = Find .A B′∩

Solution:

14 Unit 1: Sets

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 A B

1 2

3

4

Unit 1: Sets 15

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. ( )A B ′∪

F. ( )A B ′∩

G. A B′ ′∪

H. A B′ ′∩

A.

B.

C.

D.

E.

F.

G.

H.

16 Unit 1: Sets

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 A B

1 2

3

4

Unit 1: Sets 17

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: A B− does not mean the same thing as .B A−

18 Unit 1: Sets

Shading Venn Diagrams

Use the numbered regions to help you determine which regions to shade. Think of it as “paint-

by-number.”

Example 12: In the Venn diagram below, shade the region(s) corresponding to .A B′∩

Solution:

U A B

1 2

3

4

Unit 1: Sets 19

Example 13: In the Venn diagram below, shade the region(s) corresponding to ( ) .A B C′∪ ∩

Solution:

U

A B

C

20 Unit 1: Sets

§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∪ if ( ) ( )19, 14,n A n B= = and ( ) 4.n A B∩ =

Solution:

Unit 1: Sets 21

Developing a Formula for ( )n A B∪

In the above Venn diagram,

• Determine the regions accounted for in each of the following.

o ( )n A counts the elements in regions ____________

o ( )n B counts the elements in regions ____________

o ( ) ( )n A n B+ counts the elements in regions ____________ and ________

o ( )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∪ =

U A B

1 2

3

4

22 Unit 1: Sets

Example 3: Find the value of ( )n A B∩ if ( ) ( )30, 19,n A n B= = and ( ) 43.n A B∪ =

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.

( ) ( ) ( ) ( )21, 6, 26, 7,n A B n A B C n A C n B C∩ = ∩ ∩ = ∩ = ∩ = ( ) ( ) ( ) ( )20, 25, 40, and 2.n A C n B C n C n A B C′ ′ ′ ′ ′∩ = ∩ = = ∩ ∩ =

U

A B

C

Unit 1: Sets 23

Example 5: A survey of 80 movie renters was taken. 25 enjoy horror films. 45 enjoy romantic comedies. 32 enjoy documentaries. 10 enjoy horror films and romantic comedies. 12 enjoy romantic comedies and documentaries. 7 enjoy horror films and documentaries. 4 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