I've just began reading Walter Rudin's "Principles of Mathematical Analysis". Early on in the text is introduced subset notation, with which I am familiar. My problem is that Rudin doesn't seem to differentiate between the notation for "is a subset of" ( $\subseteq$ ) and "is a proper subset of" ( $\subset$ ).
I'm still on chapter 1, but Rudin has thus far only used $\subset$, as in
"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 [...]"
where Rudin defines "subset". Directly after that, Rudin defines "proper subset", but does not provide notation for it:
"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."
It seems to me that differentiating between $\subseteq$ and $\subset$ is important.
Am I missing something?
Rudin's book uses ⊂ to mean subset, which is not all that uncommon. The topic of Rudin's book is not set theory and the distinction between a subset and a proper subset is often unimportant in this text, so Rudin does not introduce notation for proper subset. If needed, something like "A ⊂ B proper" may be written to specify a proper subset with this notation.