What's the use of having both a proper subset and subset definition when we have the of notion equality?

By definition a set $$A$$ is a proper subset of another set $$B$$ if we have $$A$$ is a subset of $$B$$ and $$A\neq B$$, I understand the need for such a definition however what I don't understand what's the use of the subset definition when we can say the A is a proper subset of $$B$$ in case they are not equal and $$A=B$$ in case they care, what does the subset definition really add to the table here?

• Not sure what your point is. There are situations in which we know that $A$ is a subset of $B$ but we aren't sure whether $A=B$ or not. Accordingly we may want to distinguish between cases in which we know $A$ is a proper subset and cases in which we don't. Does that answer your question?
– lulu
Dec 26, 2020 at 14:05
• what does $a\le b$ add to the table when we already have $a < b$ and $a=b$? Dec 26, 2020 at 14:19

"$$A$$ is a subset of $$B$$" is equivalent to "$$A$$ is a proper subset of $$B$$ or $$A=B$$" and the first is clearly more succint. We often don't care if a subset is proper.
• It is useful to have the $$\leq$$ relation defined on numbers; it conveys the idea that one number is at most as big as another one.
• But , arguably, numbers are sets . For example $$0=_{df} \emptyset$$ and $$1=_{df}\{\emptyset\}= \{0\}$$ , and $$2=_{df}\{0,1\}$$
• So how can we define the $$\leq$$ relation on sets? For example suppose that sets N and M are natural numbers. How are we going to express the fact that $$N\leq M$$ ?
• We will say that $$N\leq M \iff N\subseteq M$$.