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Let be $X$ an integer set: $X=\{0,1,2,\ldots,63\}$. Let be $(x,y)$ two elements from $X$ ($(x,y)\in X \times X$). I want to know if exist two transforms $T_1 :X \times X \to X$ and $T_2 :X \times X \to X$ and the following condition is met:

$T_1(x,y) =T_2(x,y)$ , but not for all values from $X \times X$.

I want to know what part of the mathematics deals with these problems.

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If $x$ is in $X$ then $(x,y)$ isn't in $X$, it's in $X\times X$. Can you edit your question so it reflects what you actually mean, please? – Gerry Myerson Jan 12 '13 at 19:29
Among other TeX improvements, I changed $X->X$ to $X\to X$. That is standard usage. – Michael Hardy Jan 12 '13 at 20:00

Your condition comes down to asking for $T_1(x,x)=T_2(x,x)$ for all $x$ --- there seems to be no condition on $T_i(x,y)$ for $x\ne y$. So for each $x$, define $T_1(x,x)$ to be anything you like, then define $T_2(x,x)$ to be equal to $T_1(x,x)$, and then define $T_i(x,y)$ any way you like for $x\ne y$.

It seems to me that despite the tags there is no linear algebra nor any modular arithmetic here. Perhaps you have left something out.

EDIT: The problem has been edited into something completely different, but not at all clear.
"$T_1(x,y)=T_2(x,y)$" is usually taken to mean equality for all pairs $(x,y)$ in $X\times X$, but you write "$T_1(x,y)=T_2(x,y)$, but not for all values from $X\times X$." So it's not at all clear for which values you want to have $T_1(x,y)=T_2(x,y)$. Please think about what you really mean, and write something that I can comprehend.

Also, I still see no justification for the tags used.

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Thanks for reply. I modified the questions to be more clear. – Ion Caciula Jan 13 '13 at 5:05

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