Give examples of two functions f: N → Z and g:
Question:

Give examples of two functions $t: \mathbf{N} \rightarrow \mathbf{Z}$ and $g: \mathbf{Z} \rightarrow \mathbf{Z}$ such that $g \circ f$ is injective but $g$ is not injective.

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

 

Solution:

Define $f: \mathbf{N} \rightarrow \mathbf{Z}$ as $f(x)=x$ and $g: \mathbf{Z} \rightarrow \mathbf{Z}$ as $g(x)=|x|$.

We first show that g is not injective.

It can be observed that:

$g(-1)=|-1|=1$

$g(1)=|1|=1$

$\therefore g(-1)=g(1)$, but $-1 \neq 1 .$

$\therefore g$ is not injective.

Now, gof. $\mathbf{N} \rightarrow \mathbf{Z}$ is defined as $g \circ f(x)=g(f(x))=g(x)=|x|$

Let $x, y \in \mathbf{N}$ such that $g \circ f(x)=g \circ f(y)$.

$\Rightarrow|x|=|y|$

Since x and y ∈ N, both are positive.

$\therefore|x|=|y| \Rightarrow x=y$

Hence, gof is injective

 

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