TITLE: Why differentiate between subset and proper subset
QUESTION [8 upvotes]: 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.

REPLY [2 votes]: For a specific example, take $B=\{1,2\}$  Then if $A$ is a proper subset, $A$ might be $\{1\}$ or $\{2\}$or $\emptyset $.  If you don't require a proper subset, $A$ might be $\{1,2\}$  Both notions are useful, so we name them.