# Is $\{ \emptyset \}$ is a subset of set $\{ \emptyset, 1, 2, 3 \}$?

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 $$\{1, 2, 3\}$$?

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

## 2 Answers

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 $$1=\{\emptyset\}$$.

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 $$S \subset E$$.

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