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I've been working through various examples of functions that are 1-1, onto or both and came across an example that I didn't really know how to tackle.

Essentially we have a function $f(x,y) = (-1)^xy$ with a domain and codomain of $Z$.

Showing that this function isn't 1-1 is relatively straight forward through the use of a counterexample (say $f(2, 1)$, $f(4, 1)$), however, I am having some trouble showing that this function is onto.

I can see from the function definition that the function is onto (as you essentially have -1 or 1 * any integer) but don't know how to show this as an actual proof.

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Just go back to the formal definition of a surjective function. Here you have a function $f : {\Bbb Z} \times {\Bbb Z} \to {\Bbb Z}$. This function is surjective if for each $z \in {\Bbb Z}$, there exists $(x,y) \in {\Bbb Z} \times {\Bbb Z}$ such that $f(x,y) = z$. Since $f(2,z) = (-1)^2z= z$, your function is indeed surjective.

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