I am beginning to read through Rudin's book, Principles of Mathematical Analysis (ed. 3) – currently at the part on the set theory definitions he uses (pp. 3). Some of these definitions are:
Definition 1: If $A$ and $B$ are sets, and if every element of $A$ is an element of $B$, we say that $A$ is a subset of $B$, and write $A \subset B$. If, in addition, there is an element of $B$ which is not in $A$, then $A$ is said to be a proper subset of $B$.
Definition 2: If $A \subset B$ and $B \subset A$, we write $A=B$. Otherwise $A \ne B$.
Based upon these definitions, it seems needless to differentiate between subset and proper subset. Take the following conjecture.
Conjecture: Suppose $A \subset B$, and $A$ is not a proper subset of $B$. Then $A=B$
Proof: If $A$ is not a proper subset of $B$, then taking the converse of Definition 1 on proper subsets, there does not exist an element of $B$ that does not belong to $A$. However, that means that every element of $B$ must belog to $A$, which according to Definition 1 on subsets, $B \subset A$ and according Definition 2, $A=B$
Following what I had worked out above, what I am wondering is – Why differentiate between subset and proper subset? Given the fact that equality if provided by Definition 2, it seems to me that the retaining a concept of "proper subset" is frivolous.
Best Answer
I don't think you'll find much traction in the mathematical crowd for eliminating the notion of proper subset.
For example, the reals are great, and all, but sometimes we're just interested in the rationals, or the naturals, or a bounded interval, or a finite subset...all of which are proper subsets of the reals.
The general idea is that a subset is comprised of some portion of the set (possibly all of it). A proper subset has a portion of the set removed. Honestly, though, I see that Brian has really encapsulated things with his comment. To be more precise, given any partial order relation (such as the subset relation), we can obtain a strict partial order relation by simply requiring irreflexivity--and given a strict partial order relation (such as the proper subset relation), we can obtain a partial order relation by allowing reflexivity.