Universal Sets and Complements

After discussing sets and power sets, we should have a good intuitive sense that some sets are a small part of a larger set.

Consider the following scenario. You are a class teacher assigning students into teams and to organize, you are putting student names into sets. Let's say you made the following three sets.

These are nine elements divided into three sets. But where were they before you made these sets? The fact is that there was a set of all students that you are borrowing these names from.

In mathematics, when we are putting elements into their respective sets, those elements often belong to some bigger and master set that we call, Universal set.

Universal Set

Universal Set is any set whose elements you are using to make other sets. If you are making sets of groups of students, the set of all students will be the Universal set. If you are putting together pairs of integers, the set of all integers is your universal set. If you are grouping movies into sets based on their actors, the set of all movies is your universal set.

Universal set is what you define and the only condition is that all children sets can only contain elements that are also present in the universal set.

Universal sets are usually denoted by $U$.

Complement of a Set

Complement of a set $A$ is usually written as $A'$ or $A^c$.

Consider the universal set $U = \{\text{Alice}, \text{Bob}, \text{Charlie}, \text{Dave}, \text{Eli}, \text{Finn}, \text{Gina}, \text{Hugh}, \text{Iris} \}$ and $A = \{\text{Alice}, \text{Bob}, \text{Charlie} \}$. The complement of set $A$ is all the elements in the universal set except the elements in set $A$.

In our case:

Complement of an Empty Set

Complement of the empty set are all the elements of the universal set that are not in the empty set, which is all of them. Thus, the complement of an empty set is the Universal set.

Complement of a Universal Set

Vice versa, complement of the universal set are all the elements of the universal set that are not in the universal set. Which is none of them. Thus the complement of a universal set is the empty set.

Complement of a Complement of a Set

It should also be intuitive that;

Symbolic Representation of Complement

To practice our set builder notation, we can write:

This is read as "$x$ such that $x$ is in the Universal Set and $x$ is not in $A$.

Visualizing Complements using Venn Diagrams

Bringing back Venn Diagrams, we can visualize the complement like this.

Tip: Use the buttons to see how complements carve out regions of U.

Complement Exercise - Find the required set based on the given universal set