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Supplemental Math: Study Aid1 chapters | 19 lessons

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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.

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 {â€¦, -3, -2, -1, 0, 1, 2, 3, â€¦} is the universal set. The answer, which is {-2, -1, 0, 1}, is a **subset** of the universal 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 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.

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.

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* = {â€¦, -3, -2, -1, 0, 1, 2, 3, â€¦}. Let's then call set *G* the set of natural numbers: *G* = {1, 2, 3, 4, â€¦}.

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

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).'

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.

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*.

We can create some expressions and equations that also represent complements of a set. For instance, if *A* is a subset of *U*, then the complement of *A* can be expressed as *U* - *A*.

For our next example, we need to establish the fact that any set is a subset of itself. In other words, we can say that a universal set *U* is a subset of itself. What then is the complement of *U* or *U* - *U*?

Well, it would be a set containing no elements, or the empty set. We need to be careful here. Because we are talking about sets, the answer is not 0. Instead, we have a set with no elements. We can show the empty set by using the symbol Ø or by using a pair of braces, { }.

If a set *A* is a **subset** of a **universal set**, then all of the elements not in *A* are the elements of the **complement** of *A*. All sets have a complement. A set and its complement never share common elements. We can look at it in another way. If we combine all the elements of a set with all of the elements of its complement, we will have created the universal set from which the original set was derived.

Once you are finished with this lesson you should be able to:

- Define the subset and complement of the subset of a universal set
- Determine the complement of a subset
- Recall the correct notation used in writing a subset and its complement

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Supplemental Math: Study Aid1 chapters | 19 lessons

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