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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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    The first part is almost identical with this question: http://math.stackexchange.com/questions/31889/mathematical-symbol-to-reference-the-i-th-item-in-a-tuple2011-05-20

1 Answers 1

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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'.