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I have a set of 2-tuples, A, and I want to state that every tuple in A has a unique second element. In other words, for every pair of elements i and j from A, where i is not equal to j, the second element of tuple i does not equal the second element of tuple j.

Also, what is the name of the logic(?) used to make such statements?

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Crossposted mathoverflow.net/questions/81606/… –  Byron Schmuland Nov 22 '11 at 13:11
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You have already made the statement perfectly well in the question, in English. Unless there is a real reason to translate it into a logical formula, natural language is the way that mathematicians normally communicate. –  Carl Mummert Nov 22 '11 at 13:11
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You could say that the projection map $\pi_2$ is injective on $A$. –  Dan Petersen Nov 22 '11 at 13:33
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1 Answer

up vote 0 down vote accepted

As Carl remarks in comments, your formulation in the question is perfectly all right. However, if you have a good reason to want to write it down symbolically (and "because that is more mathematical" is not a good reason; it is false), then you can just do it part for part:

For every pair of elements $i$ and $j$ from $A$ ($\forall i,j\in A$) where $i$ is not equal to $j$ ($i\ne j$) it holds that ($\Rightarrow)$ the second element of the pair $i$ ($\pi_2(i)$) does not equal ($\ne$) the second element of pair $j$ ($\pi_2(j)$): $$ \forall i,j\in A: i\ne j\Rightarrow \pi_2(i)\ne \pi_2(j)$$ where $\pi_2$ is the second projection function. If you don't want to use the $\pi_2$, you can write $$ \forall \langle a,b\rangle, \langle c,d\rangle\in A: \langle a,b\rangle\ne\langle c,d\rangle\Rightarrow b\ne d$$ or a bit shorter as $$ \forall \langle a,b\rangle, \langle c,d\rangle\in A: b=d\Rightarrow a=c$$ If you're in a completely formal setting where you can't quantify over patterns such as $\langle a,b\rangle$, you'll need to do something like $$ \forall a,b,x: \langle a,x\rangle\in A\land \langle b,x\rangle\in A\Rightarrow a=b$$

Since quantifiers are involved here, the logic is (first-order) predicate calculus.

By the way, the usual name for a 2-tuple, except (possibly) when you're speaking to machines instead of to people, is an ordered pair.

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Thanks for the definitions and advice, very helpful. –  sean Nov 22 '11 at 15:23
    
sorry my mistake, it's correct –  sean Nov 22 '11 at 15:36
    
offtopic: People here perfectly understand what a 2-tuple is. ;) –  beroal Nov 24 '11 at 6:52
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