Chapter 2

Sets

In this book, we will count the number of elements or objects that have a certain given property. We will constantly be answering the question Just how many elements are there like that? We will only count finite collections, but these finite collections can be difficult to count the traditional way. For example, in how many ways can you arrange six people in a row? The answer is not easily obtained by listing all of the arrangements or by simply making each of the arrangements with six people. Thus, we will have to find a different way to count these arrangements. Here is another one. How many pizzas can you make with exactly five different toppings if there are no repeated toppings and if there are ten toppings to choose from? These counting problems can become much more difficult with just a few changes in the wording. For example, consider the following counting problem. How many pizzas can you make with at most five different toppings if there are no repeated toppings and if there are ten toppings to choose from? These are typical of the counting problems that are our goal.

2.1 Set Notation

The language of mathematics is based on a grammar called Set Theory. This grammar gives us a brief and simple way to write down some very complex mathematical ideas. A set is a collection A such that given an object x, either x is in A, or x is not in A, but not both. As long as we can teil that any object is in A or not in A then A is a set. The collections we consider in this book will always be sets. The objects in A are called elements.

The elements of a set are sometimes listed between set braces, { }. For example, the following sets are described by listing the elements in them:

The first set has elements 1 and 2 but no other elements. Observe that book3 is an element in the third set but not an element of the second set. Also, Algebra is not an element of the third set unless it is book1, book2, book3, or book4.

An important set is called the empty set. This is the set that has no elements. In this book, the empty set is written in the following ways:

Thus, given an object x we know with certainty that x is not in . In terms of our counting theme, the empty set is the set with no elements, or 0 elements. For example, the set of cards in a Standard deck of 52 that are labeled by 11 make up the set {}. There are no such cards.

2.2 Predicates

The problems we are working on take place in a finite set called a universal set or a universe. This universe changes from problem to problem. For instance, if we are counting the number of red, white, and blue flags in the world, then this problem takes place in a universe of flags of the world. This set is finite. There are only a finite number of flags in this world. The set of all numbers that appear in a Standard deck of 52 cards is finite. In fact, we can list its nine elements as {2,3,4,5,6,7,8,9,10}.

Let us keep in mind that this universe that we are defining is a finite set and that it changes from problem to problem. These universes will never be the universe we live in since our surroundings are infinite once you count the abstract ideas that make up our thoughts.

The most effective means of listing a set is called a predicate. A predicate is a partial sentence that describes objects.

EXAMPLE 2.2.1 1. red, white, and blue flags is a predicate describing flags.

2. Positive even whole numbers less than 12 is a predicate describing the set {2,4,6,8,10}.

3. The predicate x2 – 4 = 0 can be used to describe the set {–2,2} as follows.

Read the symbol | as such that or as with the property that. Then x2 – 4 = 0 is a perfectly good predicate to describe the objects in a set.

4. Consider {x | x is a person on Earth}. The predicate is x is a person on Earth. This is a set that in this Computer age could be given as a finite complete list of people on the Earth, but which should not be given as that large list in this book.

5. There is also the predicate is a book, which describes {x | x is a book}.

Let A be a set in a universal set . The complement of A is the set

This is the set of elements that are not in A. This is what is outside of A in . Think of A as a circle, and then A′ is what is outside that circle.

EXAMPLE 2.2.2 Let = {0,1,2,3,4,5,6,7,8,9,10}.

1. Let A = {0,1,2,3,4,5}. Then A′ = {6,7,8,9,10}. These are the numbers in that are not in A.

2. Let B = {1,3,5,7,9}. Then B′ = {0,2,4,6,8,10}, the numbers not in B.

3. Let C be the set of even numbers in . Then C′ = {1,3,5,7,9} which is the set of odd numbers in .

4. Let be the set of children of age at most 6 years, and let B = the set of boys of age at most 6 years. Then B′ is the set of children of age at most 6 years that are not boys. These are the girls of age at most 6 years.

There are two complements that should be remembered. They come up often. We will assume the existence of a finite universal set in all that follows. Since contains everything being considered, no x satisfies x ∉ . Thus, the complement of is empty.

Since nothing is in , x ∉ for each x ∈ . Hence, the complement of is . In other words,

2.3 Subsets

When one set is contained in another set we say that the smaller is a subset of the larger. This is an intuitive way of thinking of subsets, but we will need a more precise definition of subset. Let A and B be sets. We say that B is a subset of A, and we write B ⊂ A, if

Given x ∈ B then x ∈ A.

When paraphrased, we say that B ⊂ A if each element of B is an element of A. One more way of saying B ⊂ A is that if we are given an x ∈ B then we can show that x ∈ A.

EXAMPLE 2.3.1 Let = the set of whole numbers between 0 and 25 inclusive.

1. {1,2,3} ⊂ {1,2,3,4,5}. This is True because evidently 1, 2, and 3 are elements of {1,2,3,4,5}.

2. {1,3} ⊂ {x ∈ | x is an odd number}. This is True because evidently 1 and 3 are odd numbers in .

3. Let B = the set of positive whole numbers between 1 and 25 inclusive. Each element of B is a whole number between 0 and 25 inclusive. Hence B ⊂ .

There are two examples of subsets that we should always be aware of.

Let A be a set. Evidently each element of A is an element of A. This is just mathematical double talk. Thus we conclude that

A ⊂ A.

Now consider the other end of the problem. Let A be a set. We will argue that

⊂ A.

To see this, suppose to the contrary that ⊄ A. This will lead us to a False statement. By the definition of subset, there is an element of that is not in A. But that means that has an element, which is not True of the empty set. This mathematical mistake, called a contradiction, shows us that we began with a Falsehood. You see, if we make no mistakes then a Truth leads to a Truth. The assumption that ⊄ A leads us to the untruth that contains an element. Thus it must be that we did proceed from a False premise. Hence ⊂ A.

We will have need of a list of all subsets of a set. Some counting problems use them all. Let A be a set. The set of all subsets of A is called the power set of A and it is denoted by

(A) = the set of all subsets of A.

EXAMPLE 2.3.2 Let A = {} = . Then A has exactly 0 elements, so that any subset of A has exactly 0 elements. The only set with 0 elements is . Then is the only subset of A. So (A) = {}.

EXAMPLE 2.3.3 Let A = {a}. Then A has exactly one element so that every subset of A will have at most 1 element. That is, they will have exactly 1 element or no elements. These subsets are {} which has no elements, and {a} which has exactly 1 element. So (A) = {,{a}}.

EXAMPLE 2.3.4 A = {a,b}. Then A has exactly 2 elements. Its subsets will then have exactly 2, or exactly 1, or no elements at all. The subset with no elements is {}, the subsets with exactly 1 element are {a}, {b}, and the only subset with exactly 2 elements is {a, b}. So (A) = {, {a}, {b}, {a, b}}.

2.4 Union and Intersection

In this section we study several important operations on sets. The union of sets A and B is

A ∪ B = {x | x ∈...