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Give an example of a map$(iii)\quad which\; is\; neither\;one-one\;nor\;onto$

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

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  • 1. function $:A \to B$ is one-one if $f(x)=f(y) =>x=y\qquad x,y \in A$
  • 2.A function $f:A \to B$ into if for every $y \in B$ then exists $ x \in $ such that $f(x)=y$
(iii) Let $ f:R \to R$ defined by $f(x)=1+x^2$
Let $x_1 x_2 \in R $ such that $f(x_1)=f(x_2)$
$ 1+x_1^2=1+x_2^2$
$ x_1^2=x_2^2$
$x_1=\pm x_2$
Since $f(x_1)=f(x_2)$ does not imply $x_1=x_2$
Hence f is not one-one
Consider an element -2 in codomain R.
We see that there does not exists any $ x \in R$
f defined by $ R \to R\; f(x)=1+x^2$ is not one one and not onto



answered Mar 4, 2013 by meena.p

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