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Question
Give examples of two functions f: N → Z and g: Z → Z such that g o f is injective but gis not injective.
(Hint: Consider f(x) = x and g(x) =|x|)
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Solution
Define f: N → Z as f(x) = x and g: Z → Z as g(x) =|x|.
We first show that g is not injective.
It can be observed that:
g(−1) = `|-1| = 1`
g(1) = `|1| = 1`
∴ g(−1) = g(1), but −1 ≠ 1.
∴ g is not injective.
Now, gof: N → Z is defined as
`gof(x) = g(f(x)) = g(x) = |x|`
Let x, y ∈ N such that gof(x) = gof(y).
⇒ |x| = |y|
Since x and y ∈ N, both are positive.
`:. |x| = |y| => x = y`
Hence, gof is injective
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