# Give an example of a map$(i)\quad which \;is\; one-one\; but\; not\; onto$

Note: This is the 1st part of a  3 part question, which is split as 3 separate questions here.

Toolbox:
• A function $f: X \rightarrow Y$ where for every $x1, x2 \in X, f(x1) = f(x2) \Rightarrow x1 = x2$ is called a one-one or injective function
• A function$f : X \rightarrow Y$ is said to be onto or surjective, if every element of Y is the image of some element of X under f, i.e., for every $y \in Y$, there exists an element x in X such that $f(x) = y$.
(i) Let $f:N \to N$ given by $f(x)=x^2$ we see that for $x,y \in N$
Step1: Injective or One-One function:
$f(x)=f(y)$
$=>x^2=y^2$
We take only possitive values for x and y since the function is defined in N
$x=y \qquad x,y \in N$
f is one-one
Step 2: Surjective or On-to function:
Now $2 \in N$ but there does not exist any $x \in N$ such that $f(x)=x^2=2$
since if $x^2 =2$ we see that squareroot of 2 does not belong to N
Therefore f is not onto
Solution:Hence f is one-one but not onto

edited Mar 27, 2013