# Subset of a set

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

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 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:

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 show below to avoid being redundant.

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

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