Q:

Let R be a relation and let A = $\cup\cup$R. Then R $\subseteq$ A $\times$ A. proof and notation confusion.

I'm currently reading Set Theory a First Course, and one of the lemmas is as follows.
Let R be a relation and let A = $\cup\cup$R. Then R $\subseteq$ A $\times$ A.
I'm having trouble understanding the proof. What is the difference between $\cup$R and $\cup\cup$R? Why is it important that the double union is used? Is this notation the same as R$\cup$R$\cup$R?

A:

This is due to simply definition.
Note that a relation is a set of tuples, $\cup A=\{a:a\in S$ for some $S\in A\}$ and a tuple $(a,b)$ is defined as $\{\{a\},\{a,b\}\}$.
Let $A=\cup\cup R$ and $(a,b)\in R$.
Note that, by definition,  $\forall x, x\in\cup\cup R $ iff $x\in S$ for some $S\in \cup R$ iff $x\in S$ and $S\in S'$ for some $S\in S'$ and $S\in R$.
We have $a,b\in \{a,b\}$ and $\{a,b\}\in(a,b)$ for some $\{a,b\}\in(a,b)$ and $(a,b)\in R$. Therefore $a,b\in \cup\cup R=A$, and $(a,b)\in A\times A$.
The proof as above. I can give you some example for you easier to get the idea.
Let $R=\{(a,b)\}$. Then $\cup R=\{\{a\},\{a,b\}\}$ and $\cup\cup R=\{a,b\}$.

