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Product of injective/surjective functions

Source: Romanian TST 2001

January 16, 2011
functionpigeonhole principlenumber theoryprime numbersalgebra proposedcombinatorics

Problem Statement

a) Let f,g:Z→Zf,g:\mathbb{Z}\rightarrow\mathbb{Z} be one to one maps. Show that the function h:Z→Zh:\mathbb{Z}\rightarrow\mathbb{Z} defined by h(x)=f(x)g(x)h(x)=f(x)g(x), for all x∈Zx\in\mathbb{Z}, cannot be a surjective function.
b) Let f:Z→Zf:\mathbb{Z}\rightarrow\mathbb{Z} be a surjective function. Show that there exist surjective functions g,h:Z→Zg,h:\mathbb{Z}\rightarrow\mathbb{Z} such that f(x)=g(x)h(x)f(x)=g(x)h(x), for all x∈Zx\in\mathbb{Z}.
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