How do we annotate numbers of nested sets or infinitely nested sets?

Question

Empty set: {  }

Set containing empty set: | {  } |

Set containing set containing empty set:  [ | {   } | ]

Set containing set containing set containting empty set:  ( [ | {   } | ] )

Etc.  

How do I express increasing finite or and infinite nesting of sets in a better way?  Is there a term for it I can look up?  I can find nothing on google.  Thanks.

Answer 1

Use the curly brackets each time or give the sets names.

{{1 2 3 {e π}} {a b c}} nested sets.

A={e π}, B={1 2 3 A}, C={a b c}, X={B C}

Comments

Hi Rod, Many thanks.  That's a big help.  I have 2 brief follow ups if you don't mind (sets are new to me.)  

Based on your answer, I can write my set pattern  in which each new set encloses the last one as:  

A={ }    B={A}    C={B}   D={C}    E={D} ... etc.  (Yay!)

This group of sets would not itself be a set because their order matters.  (ie. In order for B to exist, A must first exist etc.)  

Question 1:  What is an ordered group like this called?  

Question 2:  How can I write this pattern  as X=

ie.  X={the letter before X in the ordered group of sets}  

 Many thanks for any assistance you or anyone else can offer.  :-)

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I think perhaps that when one set is included within another it may be wise and more meaningful to include the curly brackets so that, for example, A is not mistaken for the alphabetic A, so B={{A}} rather than B={A}. Notation is only useful if it conveys precisely what is meant. So the double curly brackets may convey better the inclusion of a set. But, as far as I know (and I’m no set expert!), this is just an invention rather than a convention, so there may not be an official representation for what you want. A qualified, knowledgeable college or university tutor may know.

I always think it’s a good idea to explain notation (even if it’s an invention) to avoid ambiguity or misunderstanding, unless the notation is common knowledge. For example, because most input devices can’t typographically deliver “integral of f(x)dx with respect to x between the limits a and b”, I usually explain that I will be using “∫[a,b]f(x)dx” to represent it.

If A and B are sets, A=B means the sets are identical. If A={1 2 3} and B={4 5 6}, C={{A} {B}} could in this case be written C={A∪B} or C=A∪B, the latter probably being the standard option. So using set unions may be a better and more standard way of expressing the sort of thing you are looking for in some cases.

Sorry I can’t be more helpful.

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Thanks Rod.  :-)  Yeah I really had no concept of how basic or obscure this question was so that is helpful.  I'll definitely learn some more about sets and seek out a tutor like you mentioned as I would like to get a handle on this stuff (or invent one for myself!)  Also the distinction between invention and convention is helpful.  Being fairly new to math, I assumed everthing had convention, but using invention with explained notation makes sense either when there is no convention or convention is not known.