Show that the function $f: N \rightarrow N$ given by $f(x)=2 x,$ is one-one but not onto.
Show that the function $f: N \rightarrow N$ given by $f(x)=2 x,$ is one-one but not onto.
The function $f$ is one-one, for $f\left(x_{1}\right)$ $=f\left(x_{2}\right) \Rightarrow 2 x_{1}=2 x_{2} \Rightarrow x_{1}$ $=x_{2},$ Further, $f$ is not onto, as for $1 \in N ,$ there does not exist any $x$ in $N$ such that $f(x)=2 x=1$
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