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How can I prove that the following equation is both surjective and injective (thus bijective) $$F(x,y) = (x - 2 y^2 +3, y +2 x - 4)$$ ??

Mary Star
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user143899
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  • I don't see why this question has votes to close. Clearly, the OP is going to need some guidance before he/she can get on with the business of solving his/her own question. – goblin GONE May 04 '14 at 00:14

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The terms "injective" and "surjective" don't apply to equations, they apply to functions. In general, wikipedia is a decent place to start to understand new and unfamiliar terms like these, see e.g. here.

Anyway, to prove that the function $F$ you've defined is injective, suppose we're given four numbers $x,y$ and $\overline{x},\overline{y}$. Assume also that $F(x,y)=F(\overline{x},\overline{y}).$ In other words, assume:

$$x - 2 y^2 +3 = \overline{x}-2\overline{y}^2 +3$$

$$y +2 x - 4 = \overline{y} +2\overline{x}-4$$

Now prove that $x = \overline{x}$ and $y = \overline{y}.$ (This does not look entirely easy; I think you'll have to play with the problem for a while to get the answer out.)

goblin GONE
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