Question

What is the significance of { ∅ } ?

Answer 1

as in

1.618...?

Answer 2

when i was young, my physics teacher instructed us to put on our notebooks \[\huge \text V \Phi \text I\]as our subject name

Answer 3

its the limit of the ratio of consecutive fibonacci numbers

Answer 4

Actually.... I don't know.
The professor said an empty set can be represented by either { } or \(\phi\)
Of course { \(\phi\) } is not the representation of an empty set. But what is its significance?
A set of an empty set?!

Answer 5

oh
wrong symbol

Answer 6

you mean the
null set or empty set

∅\(=\emptyset =\{\}\)

Answer 7

............. I'm sorry.....

Answer 8

it is a set without any elements

Answer 9

So, { that correct symbol } = ??

Answer 10

say set A\[A=\{0,1\}\]
sub sets are
\[\{0\},\{1\},\{1,0\},\{\}\]\[=\{0\},\{1\},\{1,0\},∅\]

Answer 11

So, A= { that symbol for empty set } = \(\emptyset\) ?

Answer 12

no A has two elements

Answer 13

No, I mean for A= { that symbol for empty set }
A = ∅ ?

Answer 14

im confused , are using my definition for A or not?

Answer 15

I'm not using your definition, I'm sorry!

Answer 16

ah ok

Answer 17

I would like to ask if a set of empty set is an empty set.

Answer 18

i think so
\[\{\{\},\{\},\}=\{\}\]

Answer 19

That mean { ∅ } = ∅
while ______
^ This is NOT an empty set?

Answer 20

\[\{∅\}=\{\{\}\}=\{\}=∅\]

Answer 21

Wow! I'm confused now!

Answer 22

Which one is correct?
(i) A set of empty set gives an empty set while the set itself is also an empty set
(ii) A set of empty set gives an empty set while the set itself is not an empty set
(iii) neither

Answer 23

im leaning towards (i)
which would suggest

\[\left\{\{\{\{\{\}\}\}\{\{\{\{\}\}\}\{\{\{\{\}\}\}\}\right\}=\{\}\]

which makes sense ,

there is only one empty set

Answer 24

@RolyPoly You have GOOD intuition (correction to what was answered here)
A set containing the empty set IS NOT , repeat NOT EMPTY.

In fact WHOLE UNIVERSE can be built starting from inclusion of an empty box INSIDE ANOTHER BOX:
the von Neumann universe, or von Neumann hierarchy of sets
http://en.wikipedia.org/wiki/Von_Neumann_universe
\[\left\{ \emptyset \right\} \neq \emptyset \]

Answer 25

@UnkleRhaukus http://en.wikipedia.org/wiki/Von_Neumann_universe

Answer 26

dammit !

Answer 27

@Mikael can you explain the flaws in my intuition because i dont get this

Answer 28

Mr Yankel' Neeman was one of the most powerful and original mathematicians and Physicists of all History of Science. Fundamentals of quantum mechanics - he was one of the founders, Fundamentals of Functional Analysis - he founded, Fundamentals of real Strong first computers in USA - he invented, Atom Bomb - he built with others. So quite a guy! (a gifted Jewish boy from Hungary, later renaming himself like some fake German baron, which he was definitely NOT)

Answer 29

Explanation:

Answer 30

Empty contains NOTHING but it is already NOT nothing. It is an empty CONTAINER

Answer 31

Sorry- the 1-st words were meant to be "EMPTY BOX"

Answer 32

Sooo now, another box that contains the empty one - of course it is NOT empty

Answer 33

how come we can not draw empty sets on venn diagrams

Answer 34

The set contains an empty set (nothing). But that empty set is already something to that set :S

Answer 35

@RolyPoly there is no fixed common name for \[\left\{ \emptyset \right\} \]

Answer 36

Von Neumann called it 1 (one)

Answer 37

@Mikael In fact, I'm just asking for its significance.

Answer 38

an empty set with cardinality one?

Answer 39

No special UNIVERSAL significance, only if one is working with Von Neumann hierarchy then IN THAT hierarchy it is special

Answer 40

Again = Imagine a box containing another box. Even if the inner box IS empty THE OUTER BOX is NOT !!

Answer 41

http://en.wikipedia.org/wiki/Von_Neumann_universe

Answer 42

hmm,
\[\boxed{\square}\]

Answer 43

As they say in the New World : YEP !

Answer 44

\[\left\{ \left\{ \right\} ,\left\{ \right\}\right\}\approx 2\]

Answer 45

approximately 2?
what?

Answer 46

There is no sign here for "CORRESPOND TO" so I arbitrary assigned this INTENDED MEANING to double wiggle

Answer 47

i dont know what you mean by corresponds to

are you saying the cardinality is 2?

Answer 48

Yes

Answer 49

Usually the term used is POWER of a set (what you called "cardinality")

Answer 50

\[\#\{\{\},\{\}\} =2\]
\[\text n\left( \{\{\},\{\}\}\right) =2\]

so it has to do with the number of commas right ?
the comma splits the set into two regions


so would this be right @mikael??
\[\#\left\{\{\{\{\{\}\}\},\{\{\{\{\}\}\},\{\{\{\{\}\}\}\}\right\}=3\]

Answer 51

In standard notation YES

Answer 52

what would Neumann call it ?

Answer 53

In his hierarchy this is not included

Answer 54

He used the set of all functions from the set to itself (power set).
But it is a kind of linguistic alphabet - according to the priority u give to
A) More brackets "around"
B) More subsets different from each other ("commas" +1)

You can create you preferred personal numbering

Answer 55

One more question...
What about { {} , {} , {} } ? What is it?

Answer 56

can you provide a better link please @Mikael ?

Answer 57

Please???? @Mikael

Answer 58

http://lileverb.blog.com/2012/07/29/stories-about-sets-read-online/ HEre it is greatly described

Answer 59

https://sites.google.com/site/ebooksabooks/Stories-about-sets-FullText-Book

Answer 60

http://math.stackexchange.com/questions/102412/proof-of-two-facts-on-the-von-neumann-hierarchy-by-transfinite-induction

Answer 61

nup

Answer 62

Is {∅} that same as [∅]? I think they are different...

Answer 63

[[∅]∅] = 2

Answer 64

yes

Answer 65

Yes, you are correct\[\varnothing \ne [\varnothing ]\]

Answer 66

Thanks @UnkleRhaukus , @Mikael and @henpen !
Even though I'm not sure what you are discussing for '' [[∅]∅] '', I can confirm that ∅≠[∅] .
Thanks again for your help!

Answer 67

@Mikael , are there any real-world applications of set theory?

Answer 68

yes money, markets, customers, google, amazon etc.

Answer 69

How are they used? Or is it too complex?

Answer 70

I must go - look up DATA MINING , Fuzzy sets