Is it possible to test $x > c$ and $y > c$ using only one condition?

If it's not possible within $(-\infty,+\infty)$, is it possible in $[0,1]$ ?


  • 1
    $\begingroup$ What do you define as "one condition"? $\endgroup$ – Alex Becker Jun 2 '12 at 21:57
  • 2
    $\begingroup$ $\min(x,y) > c$? $\endgroup$ – Henning Makholm Jun 2 '12 at 21:59
  • $\begingroup$ Please avoid "writing questions in the title". I have also reformatted the mathematical parts into $\LaTeX$ to improve readability. $\endgroup$ – Asaf Karagila Jun 2 '12 at 22:02
  • $\begingroup$ @AsafKaragila Thanks $\endgroup$ – Nison Maël Jun 2 '12 at 22:25

You want to know if $\min\{x,y\}>c$.

We can write: $$\min\{x,y\} = \frac{x+y}2 - \frac{|x-y|}2$$

Now it is simple to verify if both $x,y$ are bigger than $c$ or not.


This is a job for universality

$\rm\: a < b,c \iff a < min\{b,c\}\ $

$\rm\: a\ \ |\ \ b,c \iff a\ \ |\ \ \gcd\{b,c\}$

$\rm\: a\subset b,c \iff a\subset \ b\ \cap\ c $


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