Give examples of two functions $f: N \rightarrow Z$ and $g: Z \rightarrow Z$ such that $g \circ f$ is injective but $g$ is not injective. (Hint: Consider $f(x) = x$ and $g(x) = |x|$)
(N/A) Define $f: N \rightarrow Z$ as $f(x) = x$ and $g: Z \rightarrow Z$ as $g(x) = |x|$.
First,we show that $g$ is not injective.
We observe that $g(-1) = |-1| = 1$ and $g(1) = |1| = 1$.
Since $g(-1) = g(1)$ but $-1 \neq 1$,$g$ is not injective.
Now,$g \circ f: N \rightarrow Z$ is defined as $(g \circ f)(x) = g(f(x)) = g(x) = |x|$.
Let $x, y \in N$ such that $(g \circ f)(x) = (g \circ f)(y)$.
This implies $|x| = |y|$.
Since $x, y \in N$,both $x$ and $y$ are positive.
Therefore,$|x| = |y| \Rightarrow x = y$.
Hence,$g \circ f$ is injective.
First,we show that $g$ is not injective.
We observe that $g(-1) = |-1| = 1$ and $g(1) = |1| = 1$.
Since $g(-1) = g(1)$ but $-1 \neq 1$,$g$ is not injective.
Now,$g \circ f: N \rightarrow Z$ is defined as $(g \circ f)(x) = g(f(x)) = g(x) = |x|$.
Let $x, y \in N$ such that $(g \circ f)(x) = (g \circ f)(y)$.
This implies $|x| = |y|$.
Since $x, y \in N$,both $x$ and $y$ are positive.
Therefore,$|x| = |y| \Rightarrow x = y$.
Hence,$g \circ f$ is injective.
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