0
$\begingroup$

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,

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

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.

$\endgroup$
  • $\begingroup$ 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
0
$\begingroup$

$\emptyset$ 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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.