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,

  • $\begingroup$ $\{\emptyset\} \subset \{1,2,3,\{6,7\}\}$ would be true iff $\emptyset$ is a member of $\{1,2,3,\{6,7\}\}$. $\endgroup$ – mrs Mar 23 '16 at 13:09
  • 1
    $\begingroup$ @Clarinetist. No. $\{1\}$ and $\{\{1\}\}$ are two different sets. $\endgroup$ – almagest Mar 23 '16 at 15:11
  • $\begingroup$ 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. $\endgroup$ – DanielWainfleet Mar 25 '16 at 20:50

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.

  • $\begingroup$ 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? $\endgroup$ – Uian Mail Mar 23 '16 at 13:15
  • $\begingroup$ @UianMail : $\emptyset \neq \{ 6,7 \}$, and it's all that matter. You don't say that $3 \subset \{1,2, \{3,4\} \}$ either $\endgroup$ – Tryss Mar 23 '16 at 13:18
  • $\begingroup$ @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. $\endgroup$ – Ian Mar 23 '16 at 13:41

$\emptyset$ 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.


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