Chapter 2

Sets and Functions

Section 2.1

Sets

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 two 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}

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 = .

Sets are really representations of numbers. How do you explain to a little child what 3 is? You show them various collections of three things, desks, marbles, people, cookies. Theses collections are sets of three. We call such sets equivalent if they have the same number of elements. Equivalent sets can be put into one-to-one correspondence with each other by showing how all the elements of one set exactly match with all the elements of another set. You can represent different one-to-one correspondences by drawing arrows between them.

January

June

July

Larry

Curly

Moe

January

June

July

Larry

Curly

Moe

Can the set {January, June, July} be put into one-to-one correspondence with the set {Red, Green, Blue, Orange}?

Sets that are equal have exactly the same elements in them. Sets that are equivalent need only have the same number of elements in them.

The sets {January, June, July} and {Red, Green, Blue} are equivalent but not equal. The sets {January, June, July} and {July, June, January} are both equal and equivalent.

Picturing Sets

It 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 represent a set and you put the elements in the set inside the circle.

NO !

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.

We want to find the set

This is the complement of 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.

In the previous example we had:

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)

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

Problem Solving

How many subsets of a set are there and how many proper subsets of a set are there?

Polya Step 1 (Understand the problem)

My understanding is to be able to tell what the number of subsets and the number of proper subsets are if I know how many elements a set has.

Polya Step 2 (Devise a plan)

I will make a table with different size sets listing all the subsets and proper subsets to see if there is a pattern. I will use sets that have a small number of elements because those are easy to list.

Polya Step 3 (Carry out the plan)

Set Subsets Number of Subsets

Number of Proper

Subsets

{a} { }, {a} 2 1

{a, b} { }, {a}, {b}, {a, b} 4 3

{a, b, c} { }, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}

8 7

The number of subsets doubles each time and the number of proper subsets is 1 less than the number of subsets.

The numbers for the number of subsets form a Geometric Series in the following way:

Initial = 2 and Current = Previous 2

We get the formula that if a set has n elements the number of subsets is given by: 22(n-1)=2n.

Since the number of proper subsets is 1 less the number of proper subsets for a set with n elements is: 2n-1.

Polya Step 4 (Look Back)

How many subsets and proper subsets will the set {Red, Green, Blue, Orange, Yellow} have?

There a five elements so n=5. This give number of subsets = 25=32 and the number of proper subsets is 31.