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Complement of a Set in Math: Definition & Examples

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  • 0:02 Universal Set & Subset
  • 0:48 Complement of a Set
  • 2:16 Examples
  • 3:54 Venn Diagram
  • 4:30 Operations on Sets
  • 5:26 Lesson Summary
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Lesson Transcript
Instructor: David Liano
After completing this lesson, you will be able to define 'complement of a set' using words and using set notation. In addition, you will be able to identify a complement of a set relative to the set's respective universal set.

Universal Set and Subset

Before we define complement of a set, we should define universal set and subset because these terms will be used often in this lesson. A universal set is the set of all elements that are under consideration for a particular problem or situation.

Let's say that you are asked to find all the integers that satisfy the inequality -3 < x < 2. In this problem, the set of integers {…, -3, -2, -1, 0, 1, 2, 3, …} is the universal set. The answer, which is {-2, -1, 0, 1}, is a subset of the universal set.

Complement of a Set

Let's say that we have a set A that is a subset of some universal set U. The complement of A is the set of elements of the universal set that are not elements of A. In our example above, the complement of {-2, -1, 0, 1} is the set containing all the integers that do not satisfy the inequality.

We can illustrate this definition using a new example. If our universal set is the states of the United States, then a possible subset is the set of the New England states, which are shown here in red: A = {Connecticut, Maine, Massachusetts, New Hampshire, Rhode Island, Vermont}.

The New England States (shown in red)
new england states

The complement of A would then be the set containing all of the other states that are not part of New England. This set would contain all of the states shown in white in the accompanying map of the United States.

There are various ways to identify a set complement using notation. For instance, a prime mark can be used. Sometimes a superscripted lower case c is used, as shown here.

Figure 1
set notation

In this lesson, complement sets will be written in words, as show here. The name of the original set will have a line or underscore symbol above it.

Figure 2
set notation

Examples

Let's start off with a simple example. We will define our universal set as U = {1, 2, 3, 4, 5, 6, 7}, and we will define our subset as E = {1, 3, 4}. The complement of E is the set of all the elements in U that are not in E. Therefore, the complement of E is {2, 5, 6, 7}.

Let's now go back to the set of integers as our universal set. Therefore, our universal set is now U = {…, -3, -2, -1, 0, 1, 2, 3, …}. Let's then call set G the set of natural numbers: G = {1, 2, 3, 4, …}.

The complement of G is the set of integers that are not natural numbers. We can write this complement of G as {…, -3, -2, -1, 0}. We can also show this complement of G using another form of set notation, as show here.

Figure 3
set notation

The curvy e-symbol means 'is an element of,' and the vertical line means 'such that.' Therefore, we can read this notation as 'the complement of G is all the elements of x of the universal set (the set of integers), such that x is not an element of G (the natural numbers).'

Venn Diagram

Let's now show a graphic representation of a complement of a set using a Venn diagram. First of all, a universal set is often shown as a rectangular box. We will call A some subset of the universal set.

Figure 4
venn diagram

Here, the area inside the rectangle represents the universal set, and the white area inside the circle represents the elements of A. The blue area represents the complement of A, or the elements of the universal set that are not in A.

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