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Question

Give examples of two functions f: N ā†’ Z and g: Z ā†’ Z such that g o f is injective but g is not injective.

(Hint: Consider f(x) = x and g(x) =)

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Solution

Define f: N ā†’ Z as f(x) = x and g: Z ā†’ Z as g(x) =.

We first show that g is not injective.

It can be observed that:

g(āˆ’1) =

g(1) =

āˆ“ g(āˆ’1) = g(1), but āˆ’1 ā‰  1.

āˆ“ g is not injective.

Now, gof: N ā†’ Z is defined as.

Let x, y āˆˆ N such that gof(x) = gof(y).

ā‡’

Since x and y āˆˆ N, both are positive.

Hence, gof is injective


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