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Show that max(f(x)-g(x) , 0) ∈ C

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September 14, 2010
functionalgebra proposedalgebra

Problem Statement

Let CC be a class of functions f:N→Nf : \mathbb N \to \mathbb N that contains the functions S(x)=x+1S(x) = x + 1 and E(x)=x−[x]2E(x) = x - [\sqrt x]^2 for every x∈Nx \in \mathbb N. ([x][x] is the integer part of xx.) If CC has the property that for every f,g∈C,f+g,fg,f∘g∈Cf, g \in C, f + g, fg, f \circ g \in C, show that the function max⁡(f(x)−g(x),0)\max(f(x) - g(x), 0) is in CC, for all f;g∈Cf; g \in C.
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