What is the Difference Between Subsets and Proper Subsets?

The difference between subsets and proper subsets lies in their relationships with the sets they are subsets of.

• Subsets: A set A is a subset of set B if every element in set A is also in set B. This relationship is denoted as A ⊆ B. A subset may or may not be equal to the set it is a subset of.
• Proper Subsets: A set A is a proper subset of set B if every element in set A is also in set B, but set A does not equal set B. In other words, a proper subset contains some, but not all, of the elements of the set it is a proper subset of. This relationship is denoted as A ⊂ B.

In summary:

For example, consider sets A = {1, 2, 3} and B = {1, 2, 3, 4}. Set A is a subset of set B, but it is not a proper subset because it contains all the elements of set B. Set C = {1, 2} is a proper subset of set B because it contains some, but not all, of the elements of set B.

Comparative Table: Subsets vs Proper Subsets

The difference between subsets and proper subsets can be summarized in the following table:

A subset is a set that contains all the elements of another set, including possibly the entire set itself. On the other hand, a proper subset is a subset that is not equal to the original set; it contains some but not all elements of the original set. In other words, a proper subset contains a few elements of the original set, whereas an improper subset contains every element of the original set along with the null set.