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If every element of set $A$ is an element of set $B$ then we say that $A$ is subset of $B$. $A$ is said to be proper subset if $A$ does not equal $B$. My question is - whether the symbol we use to show subset is same as that for proper subset? I have seen that $A$ is subset of $A$ itself is shown by the same symbol as that for the subset while when we write improper subset, we usually write equal to sign with symbol of subset. So, please help me to know this.

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2 Answers 2

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I'm afraid the usage varies by author.

Some of the symbols are unambiguous: '$\subsetneq$' always means a proper subset, and '$\subseteq$' means a subset which could either be proper or not.

But the symbol '$\subset$' is ambiguous. For some, it always means a proper subset (identical to '$\subsetneq$'. For others, it just means a subset without regard to properness (identical to '$\subseteq$').

My personal preference is to use '$\subset$' to just mean any subset, proper or not. When I specifically want to exclude the possibility of equality, I explicity indicate this using '$\subsetneq$'.

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    $\begingroup$ I think the "modern way," as implied by the partial-order nature of subsets, would be to use $\subseteq$ for any subset and $\subset$ for proper subset, paralleling to $\leq$ and $<$. But I understand both preferences. $\endgroup$ Oct 12, 2015 at 16:25
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    $\begingroup$ Comment: The conscientious author will indicate the meaning of the symbol where it makes a difference. In my experience, it typically doesn't matter. $\endgroup$
    – MPW
    Oct 12, 2015 at 19:05
  • $\begingroup$ Yes, I suspect it is mostly older text, back from the days of hand-typesetting, which are more likely to abuse the notation. I can't even begin to imagine the typesetting nightmare that was Dickson's History of the Theory of Numbers. $\endgroup$ Oct 12, 2015 at 19:08
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As @MPW pointed out, $\subset$ is ambiguous. My preference, therefore, is not to use this symbol at all. I use $\subseteq$ for "subset" and $\subsetneq$ for "proper subset". So you won't encounter any ambiguities in my papers; as for other people's papers, good luck.

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    $\begingroup$ Do you refuse to publish in journals that require you to use their standard notation and use $\subset$? $\endgroup$
    – JiK
    Oct 13, 2015 at 9:12
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    $\begingroup$ @JiK I don't recall any attempt by a journal to mess up my choice of symbols. If I knew in advance that a journal will insist on $\subset$, that would certainly decrease the probability of my submitting a paper there, though it might not decrease the probability all the way to zero if there were other good reasons for wanting to publish in that particular journal (e.g., a special issue in honor of one of my friends). $\endgroup$ Oct 13, 2015 at 9:16

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