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I need to find twice continuously differentiable functions $g_i: \mathbb{R}^2 \rightarrow \mathbb{R}$ $i=1,...,I$ such that the set $\{ 0,1\}^2=\{(0,0), (1,0), (0,1), (1,1) \}$ can be written as $\{x \in \mathbb{R}^2 \text{ s.t. } g_i(x)\leq 0 \text{ } \forall i=1,...,I \}$. Could you help me?

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Hint: Try to construct a function $g$ which is $\geq0$ everywhere and has roots in $\{0,1\}^2.$

For this, first try to construct a function $h$ which is $\geq 0$ and has a root in $(0,0)$. Then, do the same for the other values in $\{0,1\}^2$. The function you are looking for is the product of the four functions you get that way.

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  • $\begingroup$ Can you be more specific? Your answer was obvious also to me. $\endgroup$ – STF Apr 15 '14 at 10:46
  • $\begingroup$ @Cris I edited my answer with a bit more detail. $\endgroup$ – 5xum Apr 15 '14 at 12:08
  • $\begingroup$ Why should it be the sum? $\endgroup$ – STF Apr 15 '14 at 13:15
  • $\begingroup$ Did I say sum? I of course meand product.... $\endgroup$ – 5xum Apr 15 '14 at 13:23

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