Introduction:

Understanding relationships between sets of values or objects is an important part of many mathematical processes. In this article, well look at the concept of a subset, and specifically how you can prove a subset relationship between two sets of objects. We will consider the basic rules of subsets and then look at a few examples and use them to illustrate the techniques for proving that one set is a subset of another.

What is a Subset?

A subset is a smaller collection of objects that is contained within a larger collection. For example, the set {1,2,3} is a subset of {1,2,3,4,5} because all of the elements of {1,2,3} are also included in {1,2,3,4,5}. In general, the notation A ⊆ B indicates that set A is a subset of set B.

Basics Rules of Subsets

There are a few key rules that can help you to determine whether one set is a subset of another. It is important to remember that a subset relationship is only valid if all of the elements of the first set are contained within the second set; if any elements are missing, then the relationship between the sets is not a subset relationship.

Here are the three key rules for determining subsets relationships:

1. Any set is a subset of itself (A ⊆ A).
2. The empty set (ω) is a subset of any set (ω ⊆ A).
3. If all elements of set A are contained in set B, then A is a subset of B (A ⊆ B).

Proving that B_A is a Subset of A

To prove that B_A is a subset of A, you must first determine whether all of the elements of B_A are included in A. If so, then A ⊆ B and B_A ⊆ A. To do this, you will have to examine both sets and use elements of B_A to construct a list of elements in A that contains all of the elements of B_A.

For example, if A = {1,2,3,4,5} and B_A = {2,3}, you could look at the elements of B_A and create the list {2,3,4}, which contains both elements of B_A and is a subset of A.

Conclusion: