Relations Between Sets

There are two basic relationships between sets: equal sets and subsets.

Equal Sets

If two sets have the same elements, we say the sets are equal, and write A = B if our sets are A and B. If they are not equal, we use the $\ne$ sign. For example A $\ne$ B.

Subsets

Another relationship between sets that is very important is the concept of a subset. We say A is a subset of B if all the elements of set A are found in set B. Symbolically, A is a subset of B would be written like this,

$A\subseteq B$

There are two notations for subset, if A is a proper subset of B, we would show it symbolically like this,

$A\subset B$

It would mean that all the elements of A are in B, but set B has at least one other element not in A.

Example 1

If A = {1, 2, 3} and B= {1, 2, 3, 4, 5} we would say A is a proper subset of B, or A⊂ B since all the elements of A are in B, but B has the elements 4 and 5 that are not in A. Sometimes it helps to remember that set B must be larger since it has at least one element more than A.

On the other hand, if A = {a, b, c} and B = {All lowercase letters from a to c} then $A\subseteq B$ but not a proper subset.

As the example above demonstrates, a set is a subset of itself, but it is not a proper subset.  If A is a subset of B and it is possible that A=B then we use the notation

$A\subseteq B$

Note the line under the subset notation helps you remember that these two sets could be equal.

Example 2

Is { } a proper or improper subset of the set B = {1, 2, 3}?

Yes, it is a proper subset because all of { } is in set B, and B contains at least one other element not in { }.  It might help to consider this:  if you take out the 1, 2, and 3, what is left? The empty set, of course! This leads to the following rule.

The number of subsets of a set is completely determined by the number of elements in the set. It has nothing to do with what the elements are. If a set A has n elements then it must have $2^n$ subsets. For example if A is a 3 element set, say A={1,2,3}, then there must be $2^3=8$ subsets of A. Let's list them to make sure. They are {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}. Similarly, if B is a set with 5 elements, there will be $2^5=32$ subsets of B.