3.1: The Basics

Learning Objectives

Students will be able to:

• Define a set using the roster method and set-builder notation
• Identify the elements of a set
• Compare and contrast the concepts of set, subset, universal set, and the empty set

Definition: Set and Elements

A set is a collection of objects or things. The objects or things in the set are called elements of the set. A set is commonly denoted by a capital letter and defined by describing the contents, or by listing the elements of the set, enclosed in braces. For example, set A, containing the elements 2, 3, and 4, can be written as A = {2, 3, 4}.

Example \(\PageIndex{1}\)

What is the name of the given set below and what are its elements?

C = {1, 6, 2, 5}

Solution

The name of the set is "C" and its elements are 1, 6, 2, and 5.

In the previous example, the set was defined by listing each element of the set enclosed in braces. This method of defining a set is called the roster method. In the next example, the set is defined by using a method called set-builder notation. Like the roster method, the elements are enclosed in braces. However, instead of listing the individual elements separated by commas as in the roster method, the set-builder notation utilizes a description of the elements. This is accomplished by first assigning a variable, usually a lower case letter, then describing the variable. Set-builder notation is often used for sets in which the number of elements is too large or impossible to write using the roster method.

Example \(\PageIndex{2}\)

What is the name of the given set below and what are its elements? Also, write the set in roster notation.

G = {x | x is a counting number less than 5}

Solution

The set of symbols, "G = {x | x is a counting number less than 5}" can be translated into English as, "The set G is the set of all x, such that x is a counting number less than 5". The name of the set is "G" and its elements are 1, 2, 3, and 4. In roster notation: G = {1, 2, 3, 4}

Let's suppose B = {y | y is a food that is tasty}. The problem with this set is that depending on personal preference, the elements of B may be different for different people. I did not think onions are "tasty" so I would not include them in B. However, my son thinks onions are "tasty" and therefore, he would incude onions in B. So, we must be sure to clearly specify a set so that in every situation, the elements of the set would be the same. This is the idea behind "well-defined".

Definition: Well-Defined

A set is well-defined if given any object, it is clear that the object is or is not an element of the set.

Example \(\PageIndex{3}\)

Determine if the given sets are well-defined

a) B = {y | y is a food that is tasty}

b) W = {w | w is a Grammy Award-winning actress prior to 2021}

c) M = { m | m is a fast car}

d) V = { v | v is a past Vice President of The United States}

Solutions

a) B is not well-defined since different people would consider different foods tasty and therefore B may contain different elements for different people.

b) W is a well-defined set because an actress was either a Grammy Award-winner or not a Grammy Award-winner prior to 2021.

c) M is not well-defined since the concept of what is "fast" for a car could vary. Is a 1968 Volkswagen Beetle fast? What is fast?

d) V is a bit tricky because depending on the time frame, a person may or may not be included in the set. We must consider that at the time we are making the determination, the list of past Vice Presidents of The United States may be determined, therefore the set is well-defined even though in the future, the elements of the set will most likely change.

Let A = { x | x is a vowel of the alphabet}. It should be clear that this set A is not well-defined because the description of the elements does not specify which alphabet the vowels are coming from (other languages have vowels) despite the fact that the set was described in English. We can use the idea of a Universal Set to help restrict the context in which the elements are considered.

Definition: Universal Set

The universal set, usually denoted by the capital letter U, is a set that establishes context. It contains all the elements within the specific context that is being examined.

Example \(\PageIndex{4}\)

Consider the set G = {x | x is a rational number less than 2} and the universe set U = {1, 2, 3, 4}. Use roster notation to rewrite set G.

Solution

G = {1}. Without the universe established the way it was, it would be quite impossible to write G using the roster notation because there is an infinite number of rational numbers less than 2. Since the universe only contained a few numbers and the number 1 was the only element of the universe that met the description in set G, we have the answer as G = {1}. Please note, changing the universe can result in different answers.

Definition: Empty Set

A set that contains no elements is called the empty set and is notated as { } or ∅.

Example \(\PageIndex{5}\)

Write the set W in roster notation when W={x ∣ x is a woman and a president of the United States}

Solution

Now at the time I write this, this set does not have any elements. So W = { } or W = Ø

Definition: Subset

A subset of a set A is another set that contains only elements from the set A, but may not contain all the elements of A.

If B is a subset of A, we write B ⊆ A.

A proper subset is a subset that is not identical to the original set – it contains fewer elements.

If B is a proper subset of A, we write B ⊂ A.

Example \(\PageIndex{6}\)

Consider these three sets:

A = the set of all even numbers, B = {2, 4, 6}, and C = {2, 3, 4, 6}

a) B ⊂ A (B is a proper subset of A) since every element of B is also an element of A (every element of B is an even number).

b) It is also true that B ⊂ C.

c) C is not a subset of A, since C contains an element, 3, that is not contained in A