Show that the function $f: N \rightarrow N$ given by $f(x) = 2x$ is one-one but not onto.

(N/A) $1$. To check if the function is one-one: Let $x_{1}, x_{2} \in N$ such that $f(x_{1}) = f(x_{2})$.
Then,$2x_{1} = 2x_{2}$,which implies $x_{1} = x_{2}$.
Since $f(x_{1}) = f(x_{2})$ leads to $x_{1} = x_{2}$,the function $f$ is one-one.
$2$. To check if the function is onto: $A$ function is onto if for every $y \in N$ (codomain),there exists an $x \in N$ (domain) such that $f(x) = y$.
Here,$f(x) = 2x$. If we take $y = 1 \in N$,then $2x = 1$,which gives $x = 1/2$.
Since $1/2 \notin N$,there is no $x \in N$ such that $f(x) = 1$.
Therefore,the function $f$ is not onto.