Can any set be a proper subset of itself & please give an example of why or why not?
As we know, a set is just a collection of objects that are similar in some way, like a gaggle of geese, pride of lions, or an army of ants.
First, let's consider the definition of a subset and a proper subset: We say that a set A is a subset of a set B if every element in A also exists in B. We say that A is a proper subset of B if A is a subset of B and there exists at least one element in B that does not exist in A.
A set cannot be a proper subset of itself.
Let A be a set. Suppose, for contradiction, that A is a proper subset of itself. By definition of proper subset, then there exists some element in A that does not exist in A. Therefore a set cannot be a proper subset of itself.
No set is a proper subset of itself because, by definition, a "proper subset" has fewer elements than the original set.