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First of all, I had no idea what to title this question, so feel free to change it to something more appropriate.

I have a set defined as such:

$\{a\frown b\mid a \in A \wedge b \in \{c\frown d\mid c \in C \wedge d \in D\}\}$

Is that logically equivalent to:

$\{a\frown (c\frown d)\mid a \in A \wedge c \in C \wedge d \in D\}$

Assuming that $\frown$ is an associative relation.

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up vote 1 down vote accepted

The sets are equal, regardless of whether $\frown$ is associative. For instance, a straightforward "element chase" shows this. More directly, note that $x=a\frown b$ for some $b$ of the form $c\frown d$ means precisely the same thing as $x=a\frown(c\frown d)$ for some $c$ and $d$.

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Awesome thanks, I have to do a proof for an assignment and I ended up moving from the one to the other and wasn't quite sure if it was a valid move to make. – DanielGibbs Sep 26 '10 at 4:08
I'm happy to help clarify, but depending on the expectations of the course and teacher, you may want to provide the element chasing argument. This would be to your own benefit, too, if the reason for equality isn't 100% clear. – Jonas Meyer Sep 26 '10 at 4:12

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