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}.
- Construct the Venn diagram to visualize these sets.
- Find Ac.
- Find A ⋃ B.
- 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.
- Find A ⋃ B
- Find A ⋂ B
- Find Ac
- 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