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 Í A

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

Now, take a look at the following Venn diagrams.

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

A = { a, b, c, f}

U = { a, b, c, f}

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

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 Ì A

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

We write B Ì A instead of B Í A

The set that contains all elements being discussed

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

Set A is not a proper subset of U because 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:

B is again a proper subset 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.

If you have any questions about the subset of a set, I will be more than happy to answer them.