4.1: Basics of Sets

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Learning Objectives

Define a set using a description, roster form or set-builder notation
Understand notations used in sets
Identify a subset of a set

An art collector might own a collection of paintings, while a music lover might keep a collection of CDs. Any collection of items can form a set. The description of the set needs to be well defined. For example, the set of "all tall students in the class" is not well defined, but "all students who are 6ft or taller" is well defined.

Set

A set is a collection of distinct objects, called elements of the set.

A set can be defined by:

describing the contents in words
by listing the elements of the set enclosed in curly brackets (known as the roster method)
by using mathematical symbols to state a rule to determine the elements of the set (known as set-builder notation)

Notation

Commonly, we will use a variable to represent a set, to make it easier to refer to that set later. We usually use capital letters to represent the set, and lower case letters for elements of a set.

Symbols used with sets:

\(\in\) means “is an element of”
\(\notin\) means “is not an element of”
| means "such that"
\(\mathbb{N}\) means the set of natural numbers, which is {1, 2, 3, 4, ... }
\(\varnothing\) means the empty set or \(\{\ \}\), which is a set that contains no elements
- Note: do not put curly brackets around \(\varnothing\). It is a symbol used to represent the empty set. \(\varnothing\) = { }. The set \( A=\{\varnothing \}\) is the set containing the empty set, A is not empty.

Example \(\PageIndex{1}\)

Some examples of sets defined by describing the contents in words:

The set of all even numbers
The set of all books written about travel to Chile

Some examples of sets defined by listing the elements of the set:

A = {1, 3, 9, 12}
B = {red, orange, yellow, green, blue, indigo, violet}

Some examples of sets defined by set-builder notation:

\( C = \{ x | x \in \mathbb{N} \text{ and } x < 10 \} \) (which is C = {1, 2, 3, 4, 5, 6, 7, 8, 9} in roster form)
\( D = \{ x | x \text{ is a day of the week that starts with S} \} \) (which is D = {Saturday, Sunday} in roster form)

A set simply specifies the contents; order is not important. The set represented by {1, 2, 3} is equivalent to the set {3, 1, 2}.

Example \(\PageIndex{2}\)

Let \(A=\{1,2,3,4\}\)

To notate that 2 is element of the set, we'd write \(2 \in A\).

Sometimes a collection might not contain all the elements of a set. For example, Chris owns three Madonna albums. While Chris’s collection is a set, we can also say it is a subset of the larger set of all Madonna albums.

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 \(B \subseteq A\).

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 \(B \subset A\).

Note that the empty set is a subset of every set: \(\varnothing \subseteq A\)

Also, every set is a subset of itself: \(A \subseteq A\)

Example \(\PageIndex{3}\)

Consider these three sets:

\(A=\) the set of all even numbers\(\quad B=\{2,4,6\} \quad C=\{2,3,4,6\}\)

Here \(B \subset A\) since every element of \(B\) is an even number, so every element in \(B\) is an element of \(A\).

More formally, we could say \(B \subset A\) since if \(x \in B,\) then \(x \in A\).

It is also true that \(B \subset C\).

\(C\) is not a subset of \(A\), since \(C\) contains an element, 3, that is not contained in \(A\). It is written \(C \nsubseteq A\).

Example \(\PageIndex{4}\)

Suppose a set contains the plays “Much Ado About Nothing”, “MacBeth”, and “A Midsummer’s Night Dream”. What is a larger set this might be a subset of?

Solution

There are many possible answers here. One would be the set of plays by Shakespeare. This is also a subset of the set of all plays ever written. It is also a subset of all British literature.

Example \(\PageIndex{5}\)

Consider the set \(A=\{a,b,c\} \). What are ALL possible subsets of \(A\)?

Solution

The subsets can contain between zero and three letters. They are

\( \varnothing, \,\, \{a\}, \,\, \{b\}, \,\, \{c\}, \,\, \{a, b\}, \,\, \{a, c\}, \,\, \{b, c\}, \,\, \{a, b, c\} \)

In general, if a set contains \(n\) distinct elements, one can create \(2^n\) different subsets.

Try it \(\PageIndex{1}\)

The set \(A=\{1,3,5\} .\) What is a larger set this might be a subset of?

Answer: There are several answers: The set of all odd numbers less than 10. The set of all odd numbers. The set of all integers. The set of all real numbers.