Subset of a Set

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In this lesson, you will learn

  • What a subset is, and how to write B ⊆ A
  • The difference between a subset and a proper subset
  • What the universal set is
  • How to show subsets in a Venn diagram

This lesson will explain what a subset of a set is. We will start our lesson with a definition.

What Is a Subset?

Definition

Set B is a subset of a set A if and only if every object of B is also an object of A.

We write B ⊆ A
Remember

By definition, the empty set ({ } or ∅) is a subset of every set. Every set is also a subset of itself, because all of its elements are in it.

Is { 1, 3 } a subset of { 1, 2, 3 }?

Show answer

Yes. Both 1 and 3 are elements of { 1, 2, 3 }, so { 1, 3 } ⊆ { 1, 2, 3 }.

Subsets in a Venn Diagram

Now, take a look at the following Venn diagrams.

Definition

Venn diagrams are closed circles, named after English logician John Venn, used to represent relationships between sets.

Venn diagram: rectangle U contains circle A; circle B with a, b and c is inside circle A, and f is in A outside B

B = { a, b, c }

A = { a, b, c, f }

U = { a, b, c, f }

Since all elements of B belong to A, B is a subset of A.

Proper Subsets and the Universal Set

Definition

Set B is a proper subset of set A if B is a subset of A and there exists an element in A that does not belong to B. We write B ⊂ A.

Having said that, B is a proper subset of A because f is in A, but not in B.

We write B ⊂ A instead of B ⊆ A.

Definition

Universal set: the set that contains all elements being discussed.

In our example, U, made with a big rectangle, is the universal set.

Set A is also a subset of U. Here it is not a proper subset, because every element of U (a, b, c and f) is in A, so A = U.

Worked example: List all the subsets of { 1, 2 }. Which ones are proper subsets?
  1. Start with the empty set: ∅ is a subset of every set.
  2. Take one element at a time: { 1 } and { 2 }.
  3. Take both elements: { 1, 2 }, the set itself.
  4. Every subset except { 1, 2 } leaves out an element, so those are proper subsets.

Answer: the subsets are ∅, { 1 }, { 2 } and { 1, 2 }. The proper subsets are ∅, { 1 } and { 2 }.

Common mistake

Writing B ⊂ A when B and A are the same set. If A = { 1, 2 }, then A ⊆ A is true, but A ⊂ A is false: a proper subset must leave out at least one element.

Is { a, b, c } a proper subset of { a, b, c }?

Show answer

No. It is a subset, because the two sets are equal, but it is not a proper subset. No element of the second set is left out.

Subsets When the Circles Are Not Inside Each Other

Notice that B can still be a subset of A even if the circle used to represent set B was not inside the circle used to represent A. This is illustrated below:

Venn diagram: rectangle U with e and f outside the circles; circle A contains a, b, c and d; a separate circle B contains c and d

As you can see, B is still a subset of A because all its objects or elements (c and d) are also objects or elements of A.

B is again a proper subset of A because there are elements of A that do not belong to B.

A and B are also subsets of the universal set U, but especially proper subsets since there are elements in U that do not belong to A and B.

In general, it is better to represent the figure above as shown below to avoid being redundant.

Venn diagram: circles A and B overlap; a and b are only in A, c and d are in the overlap, and e and f are outside both circles

The area where elements c and d are located is the intersection of A and B. More on this on a different lesson!

In the last diagram, is A a proper subset of U?

Show answer

Yes. Every element of A (a, b, c, d) is in U, and U also has e and f, which are not in A.

If you have any questions about this lesson, I will be more than happy to answer them.

Summary

  • B is a subset of A, written B ⊆ A, when every element of B is also in A.
  • B is a proper subset of A, written B ⊂ A, when B ⊆ A and A has at least one element that is not in B.
  • The empty set ∅ is a subset of every set, and every set is a subset of itself.
  • The universal set U contains all elements being discussed. In a Venn diagram it is the big rectangle.
  • B can be a subset of A even when its circle is not drawn inside the circle for A. Check the elements.

Test Yourself: 10-Question Quiz

Now check what you learned. Every answer is explained, and each time you play you get new questions.

Question 1 of 10Score: 0

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