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If I have two (or more) numbers $x, y \in \mathbb{R}$ and I am interested in the logical conditions $A$ as $x=0$ and $B$ as $y=0$. I can represent logically $A \land B$ as $x^2+y^2=0$. ie $x^2+y^2=0 \iff x=0 \land y=0$. Is there an alternative function with similar properties that ideally also uses basic field operations $+$, $-$, $\times$ and perhaps $\div$?

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The condition "$x^2+y^2=0$" is phrased in terms of $\times$ and $+$: namely, it's just "$x\cdot x+ y\cdot y=0$." Even though it uses the language of exponents, that's not necessary.

The only time when you really need to use exponentiation, and can't just get away with using multiplication instead, is when your exponent isn't an integer; but that's not the case here.

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  • $\begingroup$ Yes that's correct although I am still looking for an alternative expression. $\endgroup$ – Blair Azzopardi May 21 '17 at 6:32

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