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I came across a question which goes as follows:-

Let $S$ be a set defined as follows:- $S=\{(x,y)|(x,y)\in \mathbb{R}{\times} \mathbb{R} $ where $x\notin \mathbb{Z}$ and $y\notin \mathbb{Q}\}.$ Now, my question goes as follows:-

Can $S$ be also written as $\mathbb{R} \setminus \mathbb{Z}\times \mathbb{R} \setminus \mathbb{Q}$ ?

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If by $\Bbb R\over \Bbb Z$, you mean $\Bbb R\setminus\Bbb Z=\{ x|x\in\Bbb R \text{ and }x\notin\Bbb Z\}$, then yes. –  David Mitra Feb 22 '12 at 14:08
    
@ David Mitra, yep, that's what I meant. anyways thanks a lot. –  andrew james Feb 22 '12 at 14:10
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Please, in the future, when asking a homework question, give it a descriptive title, then give it the homework tag (in addition to other relevant tags). –  Harald Hanche-Olsen Feb 22 '12 at 14:26
    
While Harald is correct, please edit this question's title. –  Asaf Karagila Feb 22 '12 at 15:53

1 Answer 1

If by $\Bbb R\over \Bbb Z$, you mean $\Bbb R\setminus\Bbb Z=\{ x|x\in\Bbb R \text{ and }x\notin\Bbb Z\}$ (and similarly for $\Bbb R\over \Bbb Q$), then yes.

This is just the definition of Cartesion product: $$ A\times B=\{\,(a,b)\,|\, a\in A, b\in B\,\} $$

So $$ (\Bbb R\setminus\Bbb Z) \times (\Bbb R\setminus\Bbb Q) =\bigl\{\,(x,y)\,|\, x\in \Bbb R\setminus\Bbb Z, y\in \Bbb R\setminus\Bbb Q\,\bigr\} =\bigl\{\,(x,y)\,|\, x,y \in \Bbb R, x\notin\Bbb Z, y\notin \Bbb Q\,\bigr\}. $$

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