Set Notation

A set can be defined by describing its contents or by listing the elements of the set, enclosed in curly brackets.   The symbol ∈ means “is an element of”.  A set that contains no elements is called the empty set and is notated ∅.

Examples

Example 1:  The set, A, of whole numbers between 1 and 4 (inclusive) can be written as: A = {1, 2, 3, 4}.  To notate that 2 is element of the set, we’d write 2 ∈ A.

 

Example 2: Suppose the universal set is U = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Let A = {1, 2, 4, 6, 9} and B = {2, 4, 6, 8}.

  1. Construct the Venn diagram to visualize these sets.
  2. Find Ac.
  3. Find A ⋃ B.
  4. Find A ⋂ B.

 

Try It

 

Example

Suppose that the universal set, U= {purple, red, green, blue, brown, yellow, orange}.  Consider the sets:

A = {red, green, blue}
B = {red, yellow, orange}
C = {red, orange, yellow, green, blue, purple}

Find each of the following sets.

  1. Find A ⋃ B
  2. Find A ⋂ B
  3. Find Ac
  4. Find Ac ⋂ C

 

Subsets

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

Subset

The set B is a subset of the set A if every element in B is also an element 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

Consider these three sets:

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

 

Here B ⊆ A since every element of B is an even number and so is an element of A.

  • More formally, we could say B ⊆ A since if x ∈ B, then x ∈ A.

 

It is also true that B ⊆ C.

 

Notice that C is not a subset of A, since C contains an element, 3, and 3 is not an element of A.

 

See more on this example in the following video.

 

Try It

 

Example

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?

 

Example

Consider the set [latex]A = \{1, 3, 5\}[/latex]. Then, [latex]A[/latex] is a subset of which of the following sets?
[latex]X = \{1, 3, 7, 5\}[/latex]
[latex]Y = \{1, 3 \}[/latex]
[latex]Z = \{1, m, n, 3, 5\}[/latex]

 

Exercises

Given the set: A = {a, b, c, d}. List all of the subsets of A