# 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. 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: 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. 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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