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I came across through the following questions, i was able to answer some but confused in the others:

$1. \varnothing \subseteq\varnothing$

$2. \varnothing \subseteq \{\varnothing\}$

$3. \varnothing \in \varnothing$

$4. \varnothing \in \{\varnothing\}$

I know the answer to just 1.

I don't know how to write the above in latex, anyone feel free to edit, and if you can comment the link of mathjax page that was on metastackexchange.

Thanks

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  • $\begingroup$ @alexqwx thanks for the edit $\endgroup$ – Shobhit Aug 30 '14 at 12:41
  • $\begingroup$ @alexqwx Now it appears to be four times the same question! I think that the edit certainly got $3.$ and $4.$ wrong (it should be $\in$). $\endgroup$ – Danu Aug 30 '14 at 12:43
  • $\begingroup$ Yep- just copy and paste $1. \varnothing \subset\varnothing$ $2. \varnothing \subset \{\varnothing\}$ $3. \varnothing \in \varnothing$ $4. \varnothing \in\{\varnothing\}$ into the question because I can't edit it again. $\endgroup$ – beep-boop Aug 30 '14 at 12:44
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    $\begingroup$ HINT: This, and other very similar questions, were asked several times before. $\endgroup$ – Asaf Karagila Aug 30 '14 at 12:51
  • $\begingroup$ @downvoter why the downvote, i didn't knew the latex $\endgroup$ – Shobhit Aug 30 '14 at 12:58
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Set $\emptyset$ does not contain any elements so:

1) no element can be found in $\emptyset$ that is not an element of $\emptyset$ so indeed $\emptyset\subseteq\emptyset$

2) no element can be found in $\emptyset$ that is not an element of $\{\emptyset\}$ so indeed $\emptyset\subseteq\{\emptyset\}$

3) $x\notin\emptyset$ is true for any $x$ so also for $x=\emptyset$

4) Set $\{\emptyset\}$ contains $\emptyset$ as element, as in general set $\{x\}$ contains $x$ as element.


edit

The four statements above are okay. If $\subset$ replaces $\subseteq$ in 1) and 2), and stands for proper subset then the adapted 1) is not true anymore, but 2) still is.

Also it should be noted that lots of authors use $\subset$ and $\subsetneq$ instead of $\subseteq$ and $\subset$ respectively. Always check on what side the author is.

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  • $\begingroup$ so only third is false rest are true $\endgroup$ – Shobhit Aug 30 '14 at 12:47
  • $\begingroup$ Yes, that is correct. $\endgroup$ – drhab Aug 30 '14 at 12:48
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    $\begingroup$ Thank you sir, very helpful $\endgroup$ – Shobhit Aug 30 '14 at 12:49
  • $\begingroup$ You are very welcome. $\endgroup$ – drhab Aug 30 '14 at 12:49
  • $\begingroup$ @drhab Why is 1 true? how can $\varnothing \subsetneq \varnothing$ be true? $\endgroup$ – beep-boop Aug 30 '14 at 12:51
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Hint: Three of the four statements are true.

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  • $\begingroup$ @Alizter: No it's not. $\endgroup$ – Jakob Werner Aug 30 '14 at 12:47
  • $\begingroup$ While correct, it is hardly a helpful answer. $\endgroup$ – Namaste Aug 30 '14 at 12:47
  • $\begingroup$ @Alizter: I disagree and stand by that three of the four statements are true. $\endgroup$ – paw88789 Aug 30 '14 at 12:47
  • $\begingroup$ @Alizter mind posting your own answer to clear things, thanks $\endgroup$ – Shobhit Aug 30 '14 at 12:50
  • $\begingroup$ Alizter, $\subset$ is not the same as $\large\subsetneq$ in all contexts. $\endgroup$ – Namaste Aug 30 '14 at 12:50

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