Question about a subset

Question

Why is this thing true $$\{\{\varnothing\}\}\subset\{\varnothing,\{\varnothing\}\}$$

I'm trying to underestand, but I need an explanation.

Do you have a definition of what it means to be a subset? What are the elements of ${{\varnothing}}$? What are the elements of ${\varnothing,{\varnothing}}$? — hmakholm left over Monica, Oct 21 '15 at 06:32
This http://math.stackexchange.com/questions/527819/element-of-subset-of-and-empty-sets/528184#528184 will help. — Michael Hoppe, Oct 21 '15 at 06:46

score 10 · Answer 1 · edited Oct 21 '15 at 15:16

Referring to the definition of the inclusion: a set $A$ is included in another set $B$ if and only if all elements of set $A$ belongs to set $B$.

Just apply it to $$A=\{\{\varnothing\}\}, \ B=\{\varnothing,\{\varnothing\}\}.$$ What are the elements of $A=\{\{\varnothing\}\}$? There is only one which is $\{\varnothing\}$. Is $\{\varnothing\}$ an element of $B$? The answer is yes.

Conclusion: $$A \subset B.$$

score 4 · Answer 2 · answered Oct 21 '15 at 06:34

4

Let $A$ be a set; then $\{ \{ A \} \} \subset \{ A, \{ A \} \}$, for $\{ A \}$ belongs to the latter class. If you understand this, take $A := \varnothing$ and your problem is solved.

answered Oct 21 '15 at 06:34

Yes

20,719

So {{ø}}c{ø,{ø}} is true? – Oct 21 '15 at 06:36
Yes; by definition it is; nothing mysterious here. – Yes Oct 21 '15 at 06:37

Steven Alexis Gregory · Answer 3 · 2015-10-21T22:58:07.427

3

If $x \in \{\{\varnothing\}\}$ then $x = \{\varnothing\}$

Since $ x \in \{\varnothing,x\}$ then $ x \in \{\varnothing,\{\varnothing\}\}$

Hence $x \in \{\{\varnothing\}\}$ implies $x \in \{\varnothing,\{\varnothing\}\}$

So $\{\{\varnothing\}\}\subseteq\{\varnothing,\{\varnothing\}\}$

edited Oct 21 '15 at 22:58

answered Oct 21 '15 at 06:58

Steven Alexis Gregory

26,769

score 2 · Answer 4 · edited Oct 21 '15 at 15:15

2

{{$\varnothing$}} is a set that has one element in it. Its element is {$\varnothing$}; a set that contains only the empty set as a member.

{$\varnothing$, {$\varnothing$}} is a set with two elements in it. The first element is the empty set. The second element is the same element above: The set {$\varnothing$}; the set that contains only the empty set as a member.

Both sets contain the element {$\varnothing$}

Every element in the first set (it has only one) is also a member of the second set (it has two). So the first set is a subset of the second.

edited Oct 21 '15 at 15:15

Matthew Leingang

26,112

answered Oct 21 '15 at 06:39

fleablood

124,253

I did underestand thanks so much you two ^^ – Oct 21 '15 at 06:42

Question about a subset

4 Answers4