Sets and Subsets

The lesson introduces the important topic of sets, a simple idea that recurs throughout the study of probability and statistics.

Set Definitions

• A set is a well-defined collection of objects.
• Each object in a set is called an element of the set.
• Two sets are equal if they have exactly the same elements in them.
• A set that contains no elements is called a null set or an empty set.
• If every element in Set A is also in Set B, then Set A is a subset of Set B.

Set Notation

• A set is usually denoted by a capital letter, such as A, B, or C.
• An element of a set is usually denoted by a small letter, such as x, y, or z.
• A set may be described by listing all of its elements enclosed in braces. For example, if Set A consists of the numbers 2, 4, 6, and 8, we may say: A = {2, 4, 6, 8}.
• The null set is denoted by {} or ∅.
• Sets may also be described by stating a rule. We could describe Set A from the previous example by stating: Set A consists of all the even single-digit positive integers.

Sets and Probability

As we learned in the previous lesson, probability is all about statistical experiments. When a researcher conducts a statistical experiment, he or she cannot know the outcome in advance. The outcome is determined by chance.

However, if the researcher can list all the possible outcomes of the experiment, it may be possible to compute the probability of a particular outcome. The list of all possible outcomes from a statistical experiment is called the sample space. And a particular outcome or collection of outcomes is called an event.

You can see that a sample space is a type of set. It is a well-defined listing of all possible outcomes from a statistical experiment. And an event in a statistical experiment is a subset of the sample space.

Set Operations

Suppose we have a sample space S defined as follows: S = {1, 2, 3, 4, 5, 6}. Within that sample space, suppose we define two subsets as follows: X = {1, 2} and Y= {2, 3, 4}.

• The union of two sets is the set of elements that belong to one or both of the two sets. Thus, if X is {1, 2} and Y is {2, 3, 4}, the union of sets X and Y is:

  X ∪ Y = {1, 2, 3, 4}

  Symbolically, the union of X and Y is denoted by X ∪ Y.

• The intersection of two sets is the set of elements that are common to both sets. Thus, if X is {1, 2} and Y is {2, 3, 4}, the intersection of sets X and Y is:

  X ∩ Y = {2}

  Symbolically, the intersection of X and Y is denoted by X ∩ Y.

• The complement of an event is the set of all elements in the sample space but not in the event. Thus, if the sample space is {1, 2, 3, 4, 5, 6}, and Y is {2, 3, 4}, the complement of set Y is:

  Y' = {1, 5, 6}

  On this website, we denote the complement of set Y as Y'. In other places, you may see the complement of set Y denoted as Y^c.

Sample Problems

1. Describe the set of vowels.

   If A is the set of vowels, then A could be described as A = {a, e, i, o, u}.

2. Describe the set of positive integers.

   Since it would be impossible to list all of the positive integers, we need to use a rule to describe this set. We might say A consists of all integers greater than zero.

3. Set A = {1, 2, 3} and Set B = {3, 2, 1}. Is Set A equal to Set B?

   Yes. Two sets are equal if they have the same elements. The order in which the elements are listed does not matter.

4. What is the set of men with four arms?

   Since all men have two arms at most, the set of men with four arms contains no elements. It is the null set (or empty set).

5. Set A = {1, 2, 3} and Set B = {1, 2, 4, 5, 6}. Is Set A a subset of Set B?

   Set A would be a subset of Set B if every element from Set A were also in Set B. However, this is not the case. The number 3 is in Set A, but not in Set B. Therefore, Set A is not a subset of Set B.