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I have defined a set L and another set R, and a set S = L x R for the Cartesian product of the two sets.

My question is quite naive. Given an element $s \in S$, is there a formal way to express its left part $l$ and right part $r$, where $s = (l,r)$?

Another question is... given a set $S' \subseteq S$, is there a formal way to express its left part $L'$ and right part $R'$, where $S'=L' \times R'$?

Thank you very much!

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One might use the notations $s_1$, $s_2$ for the first and second coordinate of $s$. Or maybe $s_0$, $s_1$. As to your $S'$ question, one would talk about the projections of $S'$ on the two axes But for general $S'$, it is almost never the case that $S'$ is the Cartesian product of its projections. –  André Nicolas May 19 '11 at 21:58
    
The first part is almost identical with this question: math.stackexchange.com/questions/31889/… –  Martin Sleziak May 20 '11 at 8:50
    
thanks all..... –  SoftTimur May 20 '11 at 11:29
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up vote 9 down vote accepted

Yes, you can use the projection maps.

For example, define a map $\pi_L: S \rightarrow L$ that takes $s=(l,r)$ to $l$, and similarly define $\pi_R: S \rightarrow R$. Then, for $S' \subseteq S$, $L' = \pi_L(S')$ and $R' = \pi_R(S')$. Note this does not guarantee $S' = L' \times R'$. Consider $S' = \{(1,1),(2,1),(1,2)\}$. Then $\pi_L(S') = \{1,2\}$ and $\pi_R(S') = \{1,2\}$, but $\pi_L(S') \times \pi_R(S') = \{(1,1),(1,2),(2,1),(2,2)\} \neq S'$.

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thank you very much... –  SoftTimur May 20 '11 at 11:28
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