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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?

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  • $\begingroup$ 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? $\endgroup$
    – lulu
    Dec 26, 2020 at 14:05
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    $\begingroup$ what does $a\le b$ add to the table when we already have $a < b$ and $a=b$? $\endgroup$
    – Jamāl
    Dec 26, 2020 at 14:19

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"$A$ 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.

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  • 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 $N\leq M \iff N\subseteq M$.

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