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So the empty set $\emptyset$ is a subset of every set.

So $\emptyset \subset \{1,2,3\}$

But why isn't

$\{\emptyset\} \subset \{1,2,3,\{6,7\}\}$

Shouldn't it be valid because {..other objects..{6,7}} contain every object present in {${\emptyset}} which is nothing.

and even if we enclose a empty set inside a set isn't it still going to create another empty set?

$\{\emptyset\} = \{\{\}\}$ is this equal to $\{\}$ since there is nothing inside.

Thanks for the help,

Uian Mail
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  • ${\emptyset} \subset {1,2,3,{6,7}}$ would be true iff $\emptyset$ is a member of ${1,2,3,{6,7}}$. – Mikasa Mar 23 '16 at 13:09
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    @Clarinetist. No. ${1}$ and ${{1}}$ are two different sets. – almagest Mar 23 '16 at 15:11
  • Suppose $x={;{};}$ . This means $\forall y;(y\in x\iff y={};)$. So if $y$ is the empty set then $ y\in x.$ So $x$ has a member ( namely, $\phi \in x$) .So $x$ is not the empty set. – DanielWainfleet Mar 25 '16 at 20:50

3 Answers3

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1) $\{ \emptyset \} \not\subset \{1,2,3,\{6,7\}\}$, because $\emptyset \not \in \{1,2,3,\{6,7\}\}$.

Indeed, $\emptyset \neq 1$ , $\emptyset \neq 2$, $\emptyset \neq 3$ and $\emptyset \neq \{6,7\}$

2) $\{\}$ could be a notation for $\emptyset$, it's unusual (= not used) and prone to missunderstanding, but it's logical.

Tryss
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  • But doesn't {6,7} contain $\emptyset$. If A is a subset of B then B contains everything present in A, so doesn't this mean everything in A belongs to B? – Uian Mail Mar 23 '16 at 13:15
  • @UianMail : $\emptyset \neq { 6,7 }$, and it's all that matter. You don't say that $3 \subset {1,2, {3,4} }$ either – Tryss Mar 23 '16 at 13:18
  • @UianMail You may be confused about how these symbols are pronounced. $\emptyset$ is contained in ${ 6,7 }$; symbolically $\emptyset \subseteq { 6,7 }$. But it is not an element of ${ 6,7 }$; symbolically $\emptyset \not \in { 6,7 }$. The empty set can be an element of a set because it is an object in its own right. – Ian Mar 23 '16 at 13:41
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$\emptyset$ is a subset of every set; not a member of every set.
$\{\emptyset\}$ is not empty; it has a member; namely $\emptyset$.
By definition, $A\subseteq B$ iff every element of $A$ is contained in $B$.
Is $\emptyset$ an element of $\{1,2,3,\{6,7\}\}$? in other words:
Is $\emptyset$ one of these elements : $1,2,3,$ or $\{6,7\}$?

Here you can see good question and answer about using "{}" in set theory.

Community
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user 1
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(1) “and even if we enclose a empty set inside a set isn't it still going to create another empty set?”
: Roughly, $∅$ means ‘a box which contains nothing’. {$∅$} means a box which contains another box which contains nothing. They are not the same. So, {$∅$} is not empty, it contains an object.
(2) "{∅}={{}} is this equal to {} since there is nothing inside."
: No, they are not equal. {$∅$} means a box B which contains an empty box A. {} means the box A which contains nothing.

Shekhar Malhotra
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