Let f : X rightarrow Y and g : Y rightarrow 2 be function. The following two par

Question

Let f : X rightarrow Y and g : Y rightarrow 2 be function. The following two parts are independent from each other. Suppose that g o f is one to one and f is onto Y. Prove that g is one to one. Construct an example to show that if g o f is one to one but f is not onto Y g does not have to be one to one

Explanation / Answer

a) If g(a)=g(b) then since f is onto, there exists elements x,y such that f(x)=a and f(y)=b so g(a)=[g o f](x) and g(b)=[g o f](y) => [g o f](x)=[g o f](y) and since g o f is 1-1, x = y. Thus f(x)=f(y) => a = b So g is 1-1. b) Let f: Z -> Z and g: Z-> Z with f: x -> 2x and g: x -> [0.5 x] (g maps to 0.5x, rounded down). Now g o f: x -> x which is clearly 1-1, but g is not 1-1 since g(1)=g(0)=0

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