Subset of a set

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

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

By definition, the empty set( { } or ∅ ) is a subset of every set.

Now, take a look at the following Venn diagrams.

Definition of Venn Diagrams:

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

Venn diagrams

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 subset:

Set B is a proper subset of set A, if 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.

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 proper subset of U because not all elements of U are in subset A.

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

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 does 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 does 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

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

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

Subset of a set quiz. Use the quiz below to see how well you can recognize subsets. 

Tough algebra word problems

100 Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius!

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