Section SV.2 – Union, Intersection, and Complement

It is very common for sets to interact. For example, you and a new roommate decide to have a house party, and you both invite your circle of friends. At this party, two sets are being combined, though it might turn out that there are some friends that were in both sets.

Union, Intersection, and Complement

The union of two sets contains all the elements contained in either set A, or set B, or in both sets. The union is written “A ⋃ B” or “A or B”.

The intersection of two sets contains only the elements that are in both sets A and B. The intersection is written “A ⋂ B” or “A and B”.

The complement of a set A contains everything that is not in the set A. The complement of A can be written Ac, A̅, A’, not A, or sometimes ~A. For consistency, this text will use A̅.

For the house party described above, the union is the set of all your friends along with the friends of your roommate or all the friends together.

The intersection will be the friends that you and your roommate share in common.

Example 1

Consider the sets:

A = {red, green, blue}

B = {red, yellow, orange}

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

D = {yellow, white}

a) Find A ⋃ B

The union contains all the elements in either set A or set B, or both:

A ⋃ B = {red, green, blue, yellow, orange}
Notice we only list red once.

b) Find A ⋂ B

The intersection contains all the elements in both sets: A ⋂ B = {red}

c) Find A̅ ⋂ C

Here we are looking for all the elements that are not in set A and are also in C.

To solve this problem, write out all the elements of set C, then remove the elements that are also contained in set A. A̅ ⋂ C = {orange, yellow, purple}

d) Find A ⋂ D.

Because set A and set D have no elements in common, we can write:

A ⋂ D = ∅ or A ⋂ D = {  }
You Try SV.2.A

Using the sets from the previous example, find A ⋃ C and B̅ ⋂ A.

Notice that in the example above, it would be hard to just ask for A̅ since everything from the color fuchsia to puppies and peanut butter are included in the complement of the set. For this reason, complements are usually only used with intersections, or when we have a universal set in place.

Universal Set

A universal set is a set that contains all the elements we are interested in.
This would have to be defined by the context of the problem.

A complement is relative to the universal set, so A̅ contains all the elements in the universal set that are not in A.
Example 2
a. If we were discussing searching for books, the universal set might be all the books in the library.

b. If we were grouping your Facebook friends, the universal set would be all your Facebook friends.

c. If you were working with sets of numbers, the universal set might be all whole numbers, all integers, or all real numbers.

d. If you were working with letters, the universal set might include all 26 letters in the English alphabet.
Example 3

Suppose the universal set is U = all whole numbers from 1 to 9.

If A = {1, 2, 4}, then A̅ = {3, 5, 6, 7, 8, 9}
You Try SV.2.B

Suppose the universal set is U = all odd numbers from 1 to 15.

If A = {1, 3, 9, 11}, then A̅ = ______________________________
Set Operations

As we saw earlier with the expression A̅ ⋂ C, set operations can be grouped together. Grouping symbols can be used like they are with arithmetic, which is to force the order of operations.

Rules for set operations:

  1. Work from left to right.
  2. Set operations within parenthesis should be done first.
  3. If there is a complement inside the parenthesis, the complement should be done first.
Example 4

Suppose H = {cat, dog, rabbit, mouse}, F = {dog, cow, duck, pig, rabbit}, W = {duck, rabbit, deer, frog, mouse}

a. Find (H ⋂ F) ⋃ W

We start with the intersection: H ⋂ F = {dog, rabbit}

Now we union that result with W: (H ⋂ F) ⋃ W = {dog, duck, rabbit, deer, frog, mouse}

b. Find H ⋂ (F ⋃ W)

We start with the union: F ⋃ W = {dog, cow, rabbit, duck, pig, deer, frog, mouse}

Now we intersect that result with H: H ⋂ (F ⋃ W) = {dog, rabbit, mouse}

c. Find [latex]\overline{(H~\cap~F)}~\cap~W[/latex]

We start with the intersection: H ⋂ F = {dog, rabbit}

Now we want to find the elements of W that are not in H ⋂ F

[latex]\overline{(H~\cap~F)}~\cap~W[/latex] = {duck, deer, frog, mouse}

Note: You normally need to know the universal set to find the complement of a set. However, in this problem we were able to determine the answer by thinking about the meaning of the set operations.

Section SV.2 – Answers to You Try Problems

SV.2.A

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

B̅ ⋂ A = {green, blue}

SV.2.B

A̅ = {5, 7, 13, 15}

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College Mathematics - MAT14X - 3rd Edition Copyright © by Adam Avilez; Shelley Ceinaturaga; and Terri D. Levine is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

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