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Question
Give examples of two functions f : N → Z and g : Z → Z, such that gof is injective but gis not injective.
Solution
Let f : N → Z be given by f (x) = x, which is injective.
(If we take f(x) = f(y), then it gives x = y)
Let g : Z → Z be given by g (x) = |x|, which is not injective.
If we take f(x) = f(y), we get:
|x| = |y|
⇒ x = ± y
Now, gof : N → Z.
(gof) (x)=g (f (x)) = g (x) = |x|
Let us take two elements x and y in the domain of gof , such that
(gof) (x) = (gof) (y)
⇒ |x| = |y|
⇒ x = y (We don't get ± here because x, y ∈N)
So, gof is injective.
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