Relations Between Sets

There are two basic relationships between sets: equal sets, and subsets. The universal set U contains all elements that you are currently considering.

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 ≠ sign for example A ≠ 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 ⊆ B

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

A ⊂ 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 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 ⊆ 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 ⊆ 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 empty set is a subset of every set.

The Universal Set U

The universal set U is the set containing all the elements you are currently considering.

Example 3

If you were working with sets of colors, the universal set would be the set containing all possible colors. If you were working with sets of planets, the universal set would be the set containing all the planets.

Does it seem logical to say that for any set A in the universal set, A ⊆ U? The answer is YES!