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I was solving the exercises in Discrete Mathematics and its applications book.

Determine whether each of these statements is true or false.

  1. {0} ⊂ {0}
  2. {∅} ⊆ {∅}

I thought both 1 and 2 are true, but when I checked the answers I found that 1 is false and 2 is true.
I got confused and distracted because I don't know the difference between them.

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    $\begingroup$ $\subset$ usually means proper subset. So if $A \subset B$, then ($a \in A \implies a \in B$) but there must be an $x \in B$ such that $x \not\in A$. Whereas $A \subseteq B$means that either $A$ is a subset of $B$ but $A$ can be equal to $B$ as well. Think of the difference between $x \leq 5$ and $x<5$. $\endgroup$ – Anurag A Oct 2 '18 at 20:53
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    $\begingroup$ In this context, $A\subset B$ means that $A$ is a proper subset of $B$, i.e., $A\neq B$ $\endgroup$ – AnotherJohnDoe Oct 2 '18 at 20:54
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    $\begingroup$ It's matter of context. In many contexts, being a proper subset is denoted with ${}\subsetneq{}$, for instance. $\endgroup$ – Bernard Oct 2 '18 at 20:57
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    $\begingroup$ It depends on the text and the notation convention the text uses. All texts will agree that $\subsetneq$ means a proper subset that is not equal. And that is false for both of these because the are equal. And all texts well agree $\subseteq$ means a subset that might or might not be equal. Ad that is true for both of these (every subset is a subset of itself). And all texts agree that $\subset$ means is a subset but they do not agree whether it means it can't be equal or whether it might be equal. Would say 1 and 2 are both true because of how i interpret $\subset$.... $\endgroup$ – fleablood Oct 2 '18 at 20:59
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    $\begingroup$ ... but it not that I am right or wrong. It's a matter of how I interpret the statements. What is right is: Both those are improper subsets that are equal. That is right. What is wrong is that 1) is a proper subset. It isn't. What matters is what is 1 saying is it saying $\{0\}$ is a proper subset of $\{0\}$ (WRONG!) or $\{0\}$ is a subset of $\{0\}$ (RIGHT). And that depends on the book. Not on any math. $\endgroup$ – fleablood Oct 2 '18 at 21:02
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If the book distinguishes between $\subset$ and $\subseteq$, then most likely the former symbol denotes proper inclusion, so $\{0\}\subset\{0\}$ is false. The latter symbol instead will denote inclusion (with possible equality).

However it's very common to find $\subset$ denoting inclusion (with possible equality), so one always has to check or try and infer from the context. Don't take Wikipedia pages as revealed truth.

It's so common that $\subset$ denotes nonstrict inclusion that somebody uses $\subsetneq$ to denote proper inclusion, for safety.

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    $\begingroup$ Personally, I avoid writing $A \subset B$ entirely, in favor of either $A \subsetneq B$ or $A \subseteq B$, for precisely this reason. $\subset$ is ambiguous, but the other two symbols are not. $\endgroup$ – Kevin Oct 2 '18 at 22:33
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    $\begingroup$ @Kevin If you look in older books you'll almost certainly find $\subset$ for nonstrict inclusion. I believe that the analogy with $<$ and $\le$ was introduced much later. I tend to avoid $\subset$, too. $\endgroup$ – egreg Oct 2 '18 at 22:46
  • $\begingroup$ @Kevin then again, there are lots of cases where it's completely unnecessary to discuss whether equality is allowed or not (e.g. if it's obvious that $B$ is not included in $A$, but you want to argue that $A$ is included in $B$), and then the $\subset$ symbol is just fine. $\endgroup$ – leftaroundabout Oct 3 '18 at 15:54
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There are two conventions and one hybrid:

You can either use

I)

i) $A \subset B$ means $A$ is a subset of $B$ and $A$ might equal $B$.

ii) $A\subsetneq C$ means $A$ is a subset of $B$ but $A \ne B$.

II)

i) $A \subseteq B$ means $A$ is a subset of $B$ and $A$ might equal $B$.

ii) $A\subset C$ means $A$ is a subset of $B$ but $A \ne B$.

Your book uses II). I personally prefer $I$ (and judging by your post you do too, it would seem). The other posters and wikipedia claim II) (which your book uses) is more common and accepted. That may be true but I'm not going to make that claim.

Anyway. If you use convention I both 1 and 2 are true. If you use convention II 1 is false and 2 is true.

(Because $\{0\}$ is a subset but is equal to $\{0\}$ and $\{\emptyset\}$ is a subset but is equal to $\{\emptyset\}$.

III) Just to avoid confusion we can do this hybrid solution that has not ambiguity:

i) $A \subseteq B$ means $A$ is a subset of $B$ and $A$ might equal $B$.

ii) $A\subsetneq C$ means $A$ is a subset of $B$ but $A \ne B$.

Just to be safe. No-one can mistake what we mean. I'd recommend using III) just to avoid this silly quibbles.

(It's actually not that bad. It's usually very clear in context which ones make sense.)

==== addendum ====

A comment points out that if the author uses two different symbols that implies the author wants the symbols to mean two different things. And by convention I) there is no symbol $\subseteq$ so it's reasonable that the author is using connvention II).

Which makes me think further about my opinion of which convention I prefer.

And I think a little concern should be made for the ease of reading and writing.

"$\subset$" means "is a subset of" and an immediate impression is on seeing it is not not really think or worry about whether it is a proper subset or not. So I believe if you want to indicate the possibility of inequality must be considered and thought about or that the possibility of equality is not an option, one should indicated this further by specifically indicating so with $\subseteq$ or $\subsetneq$. If a reader sees simply $\subset$ it is, in my opinion, not reasonable to expect the reader to consciously consider or know that equality is not possible.

So I'd go with hybrid. For reading, and trying to second guess an author I'd view in context.

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  • $\begingroup$ On "If you use convention I but 1 and 2 are true", "but" -> "both" $\endgroup$ – Pedro A Oct 3 '18 at 11:28
  • $\begingroup$ I would argue that when you use convention I), statement 1 is true and statement 2 is non-sensical because $\subseteq$ is undefined (hence when you encounter it, you should go back an check which convention the author uses). Certainly using both $\subset$ and $\subseteq$ in the same text for the same relation is confusing. $\endgroup$ – Eike Schulte Oct 4 '18 at 11:33
  • $\begingroup$ @EikeSchulte that is a good point I hadn't considered. The author in using two different symbols is implying they are different. Still... .. I will add an addendum for my new thoughts... $\endgroup$ – fleablood Oct 4 '18 at 16:05
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The book you are reading appears to follow this convention: https://en.wikipedia.org/wiki/List_of_mathematical_symbols

$A \subseteq B$ means that $A$ is a subset of $B$ (in other words, every element of $A$ is an element of $B$).

$A \subset B$ means that $A$ is a proper subset of $B$ (every element of $A$ is an element of $B$, but $A \ne B$).

However, many people use $A \subset B$ to simply mean that $A$ is a subset of $B$, not that $A$ is necessarily a proper subset of $B$.

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    $\begingroup$ The second is not a universal convention (Bourbaki, for instance). $\endgroup$ – Bernard Oct 2 '18 at 20:59
  • $\begingroup$ Fair enough, but that seems to be the meaning his book was using. $\endgroup$ – Kurt Schwanda Oct 2 '18 at 21:00
  • $\begingroup$ Wikipedia is not gospel. "but that seems to be the meaning his book was using" True. But we'd be doing the OP a disservice to imply this is universally agreed upon. $\endgroup$ – fleablood Oct 2 '18 at 21:06
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    $\begingroup$ Good points, this is not the best answer nor the accepted answer, but I made an edit to address your concerns. Thank you for the feedback! $\endgroup$ – Kurt Schwanda Oct 2 '18 at 21:12
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The strict inclusion $ A\subset B$ is used by some when $ A\subseteq B$ and $B$ has an element which is not in $A$

Thus if $A=B$ they count $ A\subset B$ as false.

On the other hand if $A=B$, both $ A\subseteq B$ and $ B\subseteq A$ are true.

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  • $\begingroup$ Much though I and many others (presumably including you) would like it to, $A\subset B$ does not universally denote strict inclusion. There are many people who use it to mean not-necessarily-strict inclusion. $\endgroup$ – David Richerby Oct 3 '18 at 15:04

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