Introduction

In this first lesson for Finite Mathematics we introduce the basic concepts and tools used in Set theory. The lesson begins with a discussion on what a set is, how it can be identified and explains the difference between finite and infinite sets. From there it moves to a discussion of the ways that two sets relate; this includes equality and other relationships. Some operations on sets are described next with a focus on "union" and "intersection". And finally the discussion turns to a pictorial representation of sets in Venn diagrams. Welcome and enjoy, as I do, the wonder of sets.

For each lesson, read the Learning Objectives and Textbook Reading Assignment first and then read the Lesson notes in ANGEL. Be sure to work through the Example problems as you read your textbook assignment. You should do the same with the examples in your online Lesson notes.

Learning Objectives

• Utilize both the roster and set-builder notation to describe a set of elements.
• Compare and contrast finite and infinite sets.
• Define the empty set and designate it with the proper notation.
• Recognize the equality of two sets.
• Recognize when one set is a subset of another set.
• Define the union and intersection of sets.
• Describe the properties of set union and intersection.
• Identify the difference between two sets.
• State the term disjoint set.
• Create Venn diagrams and use them properly.

Lesson 1 Road Map

Note: The purpose of the Lesson Road Map is to give you an idea of what will be expected of you for each lesson. Each activity in the Assignments section will be identified as individual (I), team (T), graded (G), and ungraded (U).

Readings:

• Complete the readings for Lesson 1 listed in the Course Syllabus.

Assignments:

• Complete Course Orientation (U);


• Problems: Section 6.1  1-37 odds in the textbook (U);


• Problems: Section 6.2 1-35 odds in the textbook (U);


• Lesson 1 Quiz in ANGEL (G)

Sets

A set is a well-defined collection of objects. By well-defined we mean that you are able to answer the following question: "Is a particular element in the set?" For example, let's consider the collection of names of United States presidents. We can definitely answer the question, "Is Abraham Lincoln in this collection of names of U.S. presidents?" On the other hand, suppose you have a collection of large animals. Would a horse be in this collection? Most people would say yes to that question, but suppose you asked about a collie dog? You begin to understand the meaning of well-defined here when someone asks how large is large. Large, small, good, bad, young, old -- these are all terms that are not automatically well defined and could easily cause a collection of objects not to be classified as a set.

Describing a Set

A set can be described two ways -- by roster method or by using set-builder notation.

1. Roster Method

The roster method simply lists all the elements in the set. For example, set A could be described using braces like this:

A = {1, 2, 3, 4, 5}.

We could easily tell what is in the set by just looking at it. The above is a finite set, meaning you can count the number of objects in the set.

Another version of the roster method that is used when dealing with infinite set is sometimes called the modified roster method because it uses an ellipsis " … " to show that it follows a given pattern to infinity. For example, B = {1, 2, 3, … } would be the infinite set of all positive integers.

The modified roster method can also be used on finite sets where there are many terms that follow a pattern. For example, C = {1, 2, 3,…100} would be the set of positive integers from one to one hundred. Here, the three dots show that the numbers follow the pattern set up by the first three numbers to one hundred.

Note: In set B above, we cannot list all the numbers and we cannot count them, so set B is an infinite set.

If a set is described using the roster method, you must use braces to enclose the elements.

2. Set-Builder Notation

The second method of describing a set is set-builder notation, where a set is described using the following format:

A = {x | x is an even integer > 0}

It would be read as A is the set of all x such that x is an even integer greater than zero. This set would look like the following using the roster method:

A = {2, 4, 6, …}.

The set-builder notation method can also be used to describe a set that would be cumbersome using the roster method. For example,

A = { x | x is a city in the U.S. that has more than 5,000 people living in it}.

You can imagine how many cities would be in this set!

Empty Set

Now that we know what a set is and can describe it, we need to talk about a very special set called the empty set or the null set. The more common name is empty set. It is defined as the set containing no elements and is denoted by the symbol Ø or by empty braces { }.

Relations Between Sets

There are two basic relationships between sets: equal sets, and subsets. The universal set U contains all elements that you are currently considering.

Equal Sets

If two sets have the same elements, we say the sets are equal and write A = B if our sets are A and B. If they are not equal, we use the ≠ sign for example A ≠ B.

Subsets

Another relationship between sets that is very important is the concept of a subset. We say A is a subset of B if all the elements of set A are found in set B. Symbolically, A is a subset of B would be written like this,

A ⊆ B

There are two notations for subset, if A is a proper subset of B, we would show it symbolically like this,

A ⊂ B

It would mean that all the elements of A are in B, but set B has at least one other element not in A.

Example 1

If A = {1, 2, 3} and B= {1, 2, 3, 4, 5} we would say A is a proper subset of B since all the elements of A are in B, but B has the elements 4 and 5 that are not in A. Sometimes it helps to remember that set B must be larger since it has at least one element more than A.

On the other hand, if A = {a, b, c} and B = {All lowercase letters from a to c} then A ⊆ B but not a proper subset.

As the example above demonstrates, a set is a subset of itself, but it is not a proper subset.  If A is a subset of B and it is possible that A=B then we use the notation

A ⊆ B

Note the line under the subset notation helps you remember that these two sets could be equal.

Example 2

Is { } a proper or improper subset of the set B = {1, 2, 3}?

Yes, it is a proper subset because all of { } is in set B, and B contains at least one other element not in { }.  It might help to consider this:  if you take out the 1, 2, and 3, what is left? The empty set, of course! This leads to the following rule.

The empty set is a subset of every set.

The Universal Set U

The universal set U is the set containing all the elements you are currently considering.

Example 3

If you were working with sets of colors, the universal set would be the set containing all possible colors. If you were working with sets of planets, the universal set would be the set containing all the planets.

Does it seem logical to say that for any set A in the universal set, A ⊆ U? The answer is YES!

Venn Diagrams

We need some way to discuss sets without listing all the elements in order to show certain relationships between the sets. Venn diagrams have one or more circle/s that represent set/s inside a rectangle representing the universal set.

In section 6.1 in the text, Figure 1 shows a universal set containing two sets: A, and B. The overlapping of sets A and B indicates the possibility that some elements are common to both of these sets. These common elements would be found in the overlap area.

Figure 1

Two Basic Operations on Sets

In arithmetic, we have the binary operations of addition, subtraction, multiplication, and division; in set theory we also have operations. These operations are called union of sets and intersection of sets.

Operation 1: Union of Sets

The union of two sets A and B results in another set that includes all of the elements in A together with all of the elements in B.  In other words all of the elements that are in A or in B. The symbol we use for union of sets is ∪.

Example 1

Suppose that A = {1, 2, 3} and B = {1, 3, 5, 7} then A∪B would be the set {1, 2, 3, 5, 7}.

Does it make sense to say that A∪{ } = A? If you take all of A and add nothing, what do you get? The set A, of course.

Operation 2: Intersection of Sets

The second operation on two sets is intersection. By the intersection of sets A and B we mean the set having those elements that are common to both A and B. The symbol we use for intersection is ∩.

Example 2

If we have a set A = {1, 3, 5, 7} and B = {3, 4, 5}, then A∩B  would be the set {3, 5}. Note, these are the only elements common to both sets.

Finding the intersection of sets is like sorting socks after doing a wash. You have two piles of socks from the wash and you put the matching socks together and set them in the laundry basket to distribute to the family. The paired socks in the laundry basket represent the intersection of the original two piles of socks.

Earlier, we talked about the union of a set A and the empty set resulting in set A. Can you see what the result would be if we intersect a set A with the empty set? If you think for a minute, it should be obvious: the intersection must be the empty set, since you cannot have anything in common with a set containing no elements.

Disjoint Sets and Two More Operations

Before we move on to two more operations, let's learn what "disjoint sets" are.

Disjoint Sets

Two sets A and B are called disjoint if they have no elements in common. 

In this case the intersection will be the empty set. Now look again at Figure 1 in section 6.1. Are sets A and B disjoint? This is more difficult to answer because they are showing an area of intersection for the two sets in the Venn diagram. Since this particular Venn diagram is only showing the possibility that sets A and B could have some elements in common, we cannot say for sure that A and B are disjoint or not.

Example 1

If we knew that set A contained the elements {1,2,3} and set B contained the elements {,5,6,7}, then the intersection of sets A and B would be {}, which is the empty set. So, in this case sets A and B are disjoint.

 

Operation 3: Complement of a Set

The complement of a set A, written , is the set of all those elements in the universal set that are not in A.

Example 2

If the universal set is the set {1, 2, 3, …, 10} and A = {1, 3, 5, 7, 9}, then = {2, 4, 6, 8, 10}.

If you think about it, it is apparent that any set and its complement will always be disjoint?

What is the complement of the universal set?  The empty set.

 

Operation 4: Subtraction of Sets

An operation between two sets that is not in your textbook is subtraction.

Subtraction – The set A - B (read "A minus B") is the set of all elements in A that are not in B.

Example 3

If A = {1, 2, 3, 4} and B = {1, 3, 5, 7, 9}, then A – B = {2, 4}.

Note: All we have left in the new set A – B are the elements in A that are not in B. Most people have little difficulty taking away the 1 and the 3 in this case, but sometimes wonder if they should be doing something with the 5, 7, and 9 from set B. If you remember that -- when you subtract -- you do not add anything, it will be a little easier to do.

Here is a little story I tell my classes to help them remember how to do subtraction of sets.

There is a campus party going on and several mothers find out about it and go to the party to take their daughters home. Using our example above, let set A represent the girls who were at the party and set B be the mothers who arrived to get their daughters. When the mothers arrive at the party those who had daughters there took them home, while those who did not have daughters at the party went home alone. In our example, girls 2 and 4 did not have their mothers come for them so they stayed at the party. Mothers 5, 7, and 9, even though they may have been tempted, did not stay at the party but went home, leaving only girls 2 and 4.

This story may sound a little silly to you, but if you remember it you will have very little trouble doing subtraction of sets.

Let's try a few more examples.

Example 4

If A = {a, b, c} and B = {b, c, d, e}, then A – B = {a} but B - A = {d, e}.

Example 5

Let A = {2, 5, 7, 9}, B = {1, 3, 5, 7, 8}, and C = {1, 2, 3, 4, 5, 6}.

Then, (A∪B ) –  (B∩C) = {2, 7, 8, 9}
{1, 2, 3, 5, 7, 8, 9} – {1, 3, 5} = {2, 7, 8, 9}
(A∩C) – (B∪C)={ }
{2, 5} – {1, 2, 3, 4, 5, 6, 7, 8} = Ø.

Example 6

What is the result of subtracting { } from a non-empty set D? The answer should be fairly obvious when you subtract nothing from something you get the original something so D – { } = D. In a similar sense { } – D should be { }. Remember: When you subtract D from { } you cannot add anything to { } from D; you can only take the common elements away.

Properties of Sets

De Morgan's Properties

Two more properties dealing with set operations and complements are known as De Morgan's properties.

, and

How would you go about proving these properties are true? You could pick some set to be the universal set. Next, define A and B and their complements. Then show that the properties are true. This would only show that the property holds for this specific example; we need to show that it works all the time.

To show it works for all sets we could use a Venn diagram that demonstrates all the possibilities when considering any two sets in a general universal set. Note the following Venn diagram and how its particular configuration covers all possible cases with two sets in a particular universal set.

The numbers in the following Venn diagram will be used to denote certain areas in the upcoming discussion.

Area #1 denotes the intersection of sets A and B. Areas #1 and #2 denote set A. Areas #1 and #3 denote set B. Area #4 denotes the area in the universal set not in A or B. A ∪ B would be areas [#1, #2, and #3]. A ∩ B would be area [#1]. A complement is area [#3 and #4]. B complement is area [#2 and #4].

Now let's prove the first of De Morgan's properties: .

To determine the left side of the equation we realize that A ∪ B is area [#1, #2, and #3], so the complement of those areas would be area #4. Therefore the left side gives area #4.

To determine the right side of the equation understand that A is area [#1 and #2], so the complement of A is area [#3 and #4].

B is area [#1 and #3], so the complement of B is area [#2 and #4].

The intersection of A complement and B complement is area #4. Therefore, left and right sides of the equation match and the first property is proven.

Can you prove De Morgan’s second property?  Why don’t you try it and when you think that you have it figured out, click below to see my solution.

Instructor's solution.

Note the Venn diagram, Figure 6, in section 6.1 in the textbook. In this Venn diagram we have three sets in our universal set and a total of eight areas we could use as areas to reference. Every time that a new set is added the number of different areas doubles.

Counting Elements

Number of Elements in a Set

If you wish to denote the number of elements in a set, you use the notation c(A) to denote the number of elements in set A. Putting the lowercase c in front of the set in parentheses refers to the number of elements in the set. For example: If A = (2, 4, 7, 9) we would say c(A) = 4. If we can assign a positive integer or a zero value to c(A), then set A is finite; if we cannot, then set A is infinite.

Carefully go over Example 2 in Section 6.2 in the text to see how the formula for the number of elements in the union of two sets is derived.

c(A∪B) = c(A) + c(B) - c(A∩B)

If the sets are disjoint then the last part of the formula c(A ∩ B) would be zero. Many students make mistakes by forgetting this last part of the formula and end up adding the elements in the intersection twice.

Let's look at some examples and try to pull the various ideas presented above into clearer focus.

Example 1. Sets with Numbers

Let A = {3, 4, 7, 8, 9} and B = {4, 7, 12, 15} in the universal set
{1, 2, 3, …, 15}.

If we want to find c(A∪B) , we would first have to determine A∪B = {3, 4, 7, 8, 9, 12, 15}. So c(A∪B) would then be 7.

To find c(A∪B) using the formula we would have to find the intersection A∩B= {4, 7}. Then c(A∩B) is 2.

Could we use the above formula for c(A∪B) to get the same answer? Let's try it.

c(A∪B)	= c(A) + c(B) - c(A∩B)
	=5 + 4 - 2
	=7

What about A complement? Did you get c()? Remember  ={1, 2, 5, 6, 10, 11, 12, 13, 14, 15}. So the answer should be c()=10.

Example 2. Sets with Letters

What about sets with letters instead of numbers? Are they done differently?

If A' = {a, c, d, h} and B' = {b, c, d, g}, then what is c(A' ∪B' )? Since A' ∪B'  = {a,b,c,d,g,h}, then c(A' ∪B' ) = 6. On the other hand, c(A' - B' ) = c{a,h} = 2.

Example 3. Sets with Mixed Elements.

If the universal set is {0, 1, 2, …10} and A = {0, 2, 4, 6, 8, 10} and B = {prime numbers less than 10}, find the following:

c(A∩B) = c{2} = 1      (Hint: You didn't forget what the prime numbers were, did you?)

c(A∪B) = c{0, 2, 3, 4, 5, 6, 7, 8, 10} = 9

c(∪B) = c{3, 5, 7} = 3

c( - A) = c{1, 3, 5, 7, 9} = 5

Lesson 1 Assignments

Problems in the Textbook

Please complete  this assignment in the textbook so that you will be prepared for the graded Lesson 1 Quiz. The answers to the assigned questions are in the back of the book, so you may check your answers afterwards.

• Section 6.1; Problems 1-37 odds. In other words, Problems 1, 3, 5, 7, 9, ..., 37.


• Section 6.2; Problems 1-35 odds

 

Lesson 1 Quiz (G)

Finish Lesson 1 Quiz. This link will open in a new window. You can also access this folder by selecting the Assignments link in the left side menu.

Congratulations! You have now finished Lesson 1.