# Thread: Axiom of Union (Set Theory)

1. ## Axiom of Union (Set Theory)

I have problems understanding this axiom.
It is stated:

For any set S, there exist a set B such that $\displaystyle x \in B \ iff \ x \in A \ for \ some \ A \in S.$

The wikipedia explanation doesn't really help either:

What the axiom is really saying is that, given a set S, we can find a set B whose members are precisely the members of the members of S. By the axiom of extensionality this set B is unique and it is called the union of S, and denoted ∪S.

My (infantile) understanding:
1. There is a large set S.
2. There is also a (somewhat smaller, but not necessary) set A.
3. Some elements of A also belong to S. (corollary: some elements of A lie outside of S). (is member = element?)
4. A unique set B exist in the intersection of A and S.

2. There is no general accepted or standard notation for the Zermelo-Fraenkel Axioms. This axiom is called “Axiom of the Sum Set: For any X there exists a set Y the union of all elements of X. (also called Axiom of Union)” It is sometimes symbolized as
$\displaystyle \left( {\forall X} \right)\left( {\exists Y} \right)\left( {\forall u} \right)\left[ {\left( {u \in Y} \right) \Leftrightarrow \left( {\exists z \in X \wedge u \in z} \right)} \right]$. You can find a good discussion, with examples, in ELEMENTS OF SET THEORY by H.B. Enderton.

3. Originally Posted by chopet
Tell us what book you are using.

The Axiom of Union says you can unionize all the elements together. Remember the elements are sets themselves. So if you have \{ A, B, C} then you can construct a set D which is the union of A,B,C. That is all it says.

4. Thank you for the lucid explanation. That's what I need.

I'm using Intro to Set Theory by Czech professors Karel Hrbacek and Thomas Jech.

5. Originally Posted by chopet
I'm using Intro to Set Theory by Czech professors Karel Hrbacek and Thomas Jech.
You go to CCNY?

6. Oh no.
I'm just fond of slavic mathematicians.