# Correct formal notation for sets containing sets

Informally, I have the set $A$ that contains sets of $B$ - for every element in $A$ there is a set $B$. If $n$ is the number of elements then $A_n \le B_n$. $B$ is not a subset of $A$ and $A$ is not a subset of $B$.

Abstract example that does not take into account the value types:

Imagine $A$ is a set of moms and $B$ is a set of children. Ie. "Kari's mom is Jane", "Bill's mom is Jane", "Lisa's mom is Rose", etc. Each mom will have at least one child, and may have more. Each child have one mom (lets not go into gay marriages here to make it simple :) ).

Also, $B$ can not be empty if $A$ has any objects, thus for any element in $A$ there is a set $B$. Lastly, elements in $A$ can only have real numbers, and elements in $B$ can have real numbers or natural numbers.

To describe this relationship between $A$ and $B$ formally the best notation I have come up with is:

$$A = \{x \mid x \in B, B \ne\emptyset, x =\mathbb R\}$$

$$B = \{x \mid x = \mathbb R \text{ or } x = \mathbb N \}$$

Is this the correct formal way of describing the relationship between $A$ and $B$?

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Surely not. What is $A_n, B_n$? You surely won't want $x=\mathbb R$ in the set descriptions, perhaps $x\in\mathbb R$? And also note that $\mathbb N$ is a subset of $\mathbb R$, hence "is a real or natural number" is not different from "is a real number". – Hagen von Eitzen Feb 3 '13 at 17:35
(I changed your tags for what seems to be the most appropriate one. Descriptive-set-theory is a technical area of set theory, and set-theory refers to more advanced questions. This is not about mathematical logic either, so the elementary-set-theory tag seemed the most on target.) – Andrés E. Caicedo Feb 3 '13 at 17:36
@hagen what if x must be a literal? How would one formally state that? – Inge Eivind Henriksen Feb 3 '13 at 18:42