Chapter 2

Section 2.1

Sets and Set Operations

A set is a particular type of mathematical idea that is used to categorize or group different collections of things. In mathematics the language and concept of sets are used in order to describe certain collections of things with a great amount of exactness.

A set is a collection of objects, people, numbers etc. The things in the set are called elements or members. There are three common ways to describe a set.

1. Verbal or Written description – This is often difficult to be able to describe exactly what you want (especially complicated collections of numbers).

The set of months of the year that begin with the letter J.

2. List or Roster of elements – Sets are usually named with a capital letter and the elements of the set are listed inside {} separated with a comma.

J = {January, June, July}

3. Set Builder Notation- A variable is used along with a description that evaluates to be true when the elements of the set are plugged in for the variable.

J = { x | x is a month that begins with the letter J}

The symbol is read “Is an element of”. Example January J

The symbol is “Not an element of”. Example May {January, June, July}

The symbols { } or are used to mark the empty or null set. This is a set with no elements. Example: Months that begin with the letter R = .

Picturing SetsIt is useful to be able to have a visual image of sets at times. We will make use of the Venn Diagram again to be able to draw a picture of a set or sets. Circles will make the set and you put the elements in side the circle they are in.

January

June

July

JU FebruaryMarchAprilMayAugustSeptember

OctoberNovemberDecember

The circle labeled J represents the set. The elements in the set J are inside the circle labeled J. The box labeled U represents the Universal Set. Which consists of all the months including those in J.

The universal set plays an important role in determining the complement of a set. The complement of a set are all the elements that are in the universal set but are not in the set itself. The complement of a set is marked with a bar over the letter that represents the set.

January

June

July

JU FebruaryMarchAprilMayAugustSeptember

OctoberNovemberDecember

J = {January, June, July} (The set J)

U = {January, February, March, April, May, June, July, August, September, October, November, December} (The Universal Set U)

J'= {February, March, April, May, August, September, October, November, December} (The complement of J)

J

A subset of a set is a set that makes up a portion (or all) of some other set. For example the set {June, July} is a subset of the set {January, June, July}.

January

JulyJune

FebruaryMarchAprilMayAugustSeptemberOctoberNovember

December

JU

We use the symbol to mean subset and write:

{June, July} {January, June, July}

OR

{June, July} J

The set {January, June, July} is also a subset of the set {January, June, July} (i.e. J J). This is a special case because the two sets are really equal. The set {June, July} is called a proper subset of J because it is a smaller set than J itself. We use the symbol to mark this.

{June, July} J (True) {June, July} J (True)

{January, June, July} J (True) {January, June, July} J (False)

The symbols and for sets work like < and for numbers

As we have seen before when we were talking about logic the real interesting and useful things happen when two categories of things are being discussed at once. We will give a more detailed way to show how two categories of things can interact. I want to start by going back to a previous example.

Let:

J = Months that begin with the letter J = {January, June, July}

Y = Months that end with the letter Y = {January, February, May, July}

U = Universal Set (All months) = {Jan, Feb, Mar, Apr, May, Jun, Jul, Aug, Sep, Oct, Nov, Dec}

Intersection

The intersection of two sets is marked with the symbol . It represents the elements that are common to both sets at the same time.

J Y = {January, July}

Union

The union of two sets is marked with the symbol . It represents what you get by putting together the elements in both sets (i.e. the set containing all the elements in both sets).

J Y = {January, February, May, June, July}

Y'= {June, March, April, August, September October, November, December}

J'= {May, February, March, April, August, September October, November, December}

Venn Diagrams

To represent two different sets in a Venn Diagram we put all the elements in the diagram (only once!)in the correct region for the set(s) they are in.

For example

J = {January, June, July}

Y = {January, February, May, July}

U = {January, February, March, April, May, June, July, August, September, October, November, December}

UJ Y

January

July

May

FebruaryJune

March

April

August

September

October

November

December

Various regions in the Venn Diagram can be referred to symbolically by combining the set operations intersection (), union () and complement (').

UJ Y

January

July

May

FebruaryJune

March

April

August

September

October

November

December

J ∩ (Y') = Elements in J and at the same time not in Y

= {June}

(J') ∩ Y = Elements not in J and at the same time in Y

= {May, February}

(JY)' = Elements in neither J nor Y

= {March, April, August, September, October, November, December}

As we previously mentioned each region in the Venn Diagram can be referred to using the three set operation of intersection (), union () and complement (').

UA B

BA

UA B

BA

UA B

'A

UA B

)'(BA

UA B

BA )'(

UA B

)'( BA

The number in set A is: 5 (The cardinality of A is 5. (i.e. n(A) = 5))

The number in set B is: 4 (The cardinality of B is 4. (i.e. n(B) = 4))

The number in AB is: 2 (This is how many are in both A and B. (i.e. n(AB) = 2))

The number in AB is: 7 (This is how many are in either A or B. (i.e. n(AB) = 7))

The number in A∩(B') is: 3 (This is how many are in A but not in B. (i.e. n(A(B') = 3))

The number in (AB)' is: 4(This is how many are in neither A nor B.(i.e. n((AB)')=4))

Numerical Information About Sets

We can construct the Venn Diagrams to include the number of elements in various components of sets. Rather than listing all the elements in a set we would just put in the number in that region. The number of elements in a set is called the cardinality of the set and we use the symbol n(set) to stand for it.

AU

B

dog

catmouse

ratpig

frog

snake

birdfish

cow

horse

AU

B

23 2

4

Numerical Problems with Sets

Given some numerical information about sets of things and how they are related we can extract more information based on what we know about sets.

Consider the following problem:

A survey of 100 people at the DMV asked them if they owned a car or a truck. The results were that 78 people said they owned a car and 46 people said they owned a truck. Of the 46 people who said they owned a truck 29 said they also owned a car. Answer each of the following questions:

How many own both a truck and car?

How many own either a truck or a car?

How many own just a truck?

How many own just a car?

How many own neither a truck nor a car?

How many do not own a car?

How many do not own a truck?

The idea here is to think in terms of sets and fill in the corresponding numbers in the Venn Diagram.

Let the set C = Car Owners

Let the set T = Truck Owners

C TU

2949 175

29

95

49

17

5

22

54

Numerical Information in Tables

Another way that information about the number of items in a set can be collected is in the form of a table. Lets look at the following problem.

A survey is taken of 200 students in a dorm asking if they were: male, female, democrat, republican or independent. The results are given in the table below.

Democrat

(D)

Republican

(R)

Independent

(I)

Female (F) 54 32 7

Male (M) 41 46 20

Each category is represented by a set.

D = Set of Democrats F = Set of Females

R = Set of Republicans M = Set of Males

I = Set of Independents

We can ask some more complicated questions about these categories of people.

1. How many males are there? (i.e. The number in the set M)

41 + 46 + 20 = 107

2. How many Democrats are there? (i.e. The number in the set D)

54 + 41 = 95

3. How many are both male and democrat? (i.e. The number in the set MD)

41

4. How many are either female or independent? (i.e. The number in the set FI)

54 + 32 + 7 + 20 = 113

5. How many are not republican? (i.e. The number in the set R')

54 + 41 + 7 + 20 = 122

Democrat

(D)

Republican

(R)

Independent

(I)

Female (F) 54 32 7

Male (M) 41 46 20