This page is part of a series of lessons on sets.

In these lessons, we will learn the complement of a set and the relative complement of a set.

Related Topics: More Lessons on Sets

The**complement** of set *A*, denoted by *A**’* , is the set of all elements in the universal set that are not in *A*.

Complement and Relative Complement

The complement of a set is the collection of all elements which are not members of that set. Although this operation appears to be straightforward, the way we define "all elements" can significantly change the results.
Learn what a complement of a set is.

Universal set and absolute complement.
Relative complement or difference between sets.

You can use the free Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

In these lessons, we will learn the complement of a set and the relative complement of a set.

Related Topics: More Lessons on Sets

The

The number of elements of *A* and the number of elements of *A** ’* make up the total number of elements in U .

** n( A) + n(A**

**Example: **

Let U = {*x* : *x* is an integer, –4 ≤ *x *≤ 7}, *P* = {–4, –2, 0, 2, 4, 5, 6} and

*Q ’* = {–3, –2, –1, 2, 3}.

a) List the elements of set *P ’*

b) Draw a Venn diagram to display the sets U , *P *and * P ’ *

c) Find n(*Q*)

**Solution: **

a) First, list out the members of U.

U = {–4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6, 7}

P’ = {–3, –1, 1, 3, 7} ← in U but not inP

b) Draw a Venn diagram to display the sets U , *P *and * P ’*

c) Find n(*Q*)

n( U ) = 12, n(

Q ’) = 5Use the formula:

n(

Q) + n(Q ’) = n( U )n(

Q) = n( U ) – n(Q ’) = 12 – 5 = 7

The complement of a set is the collection of all elements which are not members of that set. Although this operation appears to be straightforward, the way we define "all elements" can significantly change the results.

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.