Re: Is {{}} a valid construct?

> > suppose my universal set contains just one
> > "element" as follows:
>
> > U = { {} }
>
> Accepting this notation as far as I get it, U cannot be the Universal
> set (barring an escape signification/meaning); it is a
> set with one element only: the empty set, signified by '{}'.

According to my interpretation of Schaum, the Universal Set is any set of things under investigation (ie "in a human population study, the universal set consists of all the people in the world"). I only needed to investigate that one "element" to show a contradiction/ambiguity, but it occurs with any additional number of elements. See example in response to Celko which includes an apple.

> > What then is NOT {} ?
>
> What do you mean to say with 'NOT {}'?
> All the sets but U? Or is it 'there is no {}'(there is no empty set)?

Exactly, it is unclear.

This is how set theory works with "proper" elements: U = {apple, orange}
NOT apple = orange
NOT orange = apple
NOT {} = U This is how set theory works with "invalid" elements ( {} ): U = { {}, apple, orange}
NOT apple = {}, orange
NOT orange = {}, apple
NOT {} = apple, orange -or- {}, apple, orange

With invalid elements ( {} ), flip a coin as to which is correct for the last one. Received on Tue Feb 06 2007 - 06:02:30 CET