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Is $\{ \emptyset \}$ a subset of set $\{ \emptyset, 1, 2, 3 \}$? I know that empty set is subset of every set, but what about $\{ \emptyset \}$? What if 'right set' was just $\{1, 2, 3\}$? Will it still be true that $\{ \emptyset \}$ is a subset of set $\{1, 2, 3\}$?

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  • $\begingroup$ The set $A = \{ \emptyset, 1, 2, 3 \}$ has four elements. One of them is $\emptyset$. $\endgroup$ – Mauro ALLEGRANZA Dec 5 '18 at 15:30
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    $\begingroup$ Correct : $\emptyset$ is a subset of every set; thus $\emptyset \subseteq A$. $\endgroup$ – Mauro ALLEGRANZA Dec 5 '18 at 15:31
  • $\begingroup$ Now, the question is : is $\{ \emptyset \} \subseteq A$ ? We have to apply the def of subset ... or lists all the subsetts of $A$. $\endgroup$ – Mauro ALLEGRANZA Dec 5 '18 at 15:32
  • $\begingroup$ $\{\emptyset\}\subseteq A$ if and only if $\emptyset\in A.$ (In general $B\subseteq A$ if and only if every element of $B$ is an element of $A.$) $\endgroup$ – spaceisdarkgreen Dec 5 '18 at 15:33
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The set $\{x\}$ is a subset of $\{a,b,c\}$ if and only if $x$ is equal to $a$, $b$ or $c$. So, unless you have a weird definition of $1$, $2$ or $3$, $\{\emptyset\}\nsubseteq\{1,2,3\}$. But $\{\emptyset\}\subseteq\{\emptyset, 1,2,3\}$

Remark: The most usual construction of $\Bbb N$ from ZFC defines $0=\emptyset$ and $1=\{\emptyset\}$.

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If you are asking about $\emptyset$, then it is indeed the subset of any set.

If you are asking about $E = \{\emptyset, 1,2,3\}$, then $\emptyset \in E$ and $\emptyset \subset E$.

Moreover, if $S = \{\emptyset\}$ then $S \subset E$.

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    $\begingroup$ It looks to me like the question is asking whether $\{\varnothing\}\subseteq\{\varnothing,1,2,3\}$, which is not what you're answering here. $\endgroup$ – hmakholm left over Monica Dec 5 '18 at 15:33
  • $\begingroup$ @HenningMakholm OP changed the question twice, will redo $\endgroup$ – gt6989b Dec 5 '18 at 15:34

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