A subset refers to a set that contains some or all elements of another set, while a proper subset is a specific type of subset that contains some but not all elements of that set.
To elaborate further, if we have a set A, any set B that can be formed using elements from A is considered a subset of A. This includes the possibility of B being exactly the same as A. For instance, if A = {1, 2, 3}, then B could be {1}, {2}, {1, 2}, or even {1, 2, 3}. All these sets are valid subsets of A.
On the other hand, a proper subset is a subset that cannot be identical to the original set. This means that for set B to be a proper subset of set A, it must contain at least one element less than A. Using the previous example, if A = {1, 2, 3}, then valid proper subsets would be {1}, {2}, or {1, 2}, but not {1, 2, 3} since it is not proper—it’s identical to A.
In summary, every proper subset is a subset, but not every subset is a proper subset. The distinction lies in whether or not the set is allowed to be identical to the original set.