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  • Give an example of a map (i) which is one-one but not onto (ii) which is not one-one but onto (iii) which is neither one-one nor onto.

Give an example of a map

(i) which is one-one but not onto

(ii) which is not one-one but onto

(iii) which is neither one-one nor onto.

Answers (1)


(i) Say, f: N \rightarrow N, be a mapping defined by f (x) = x\textsuperscript{2}\\

If, f(x\textsubscript{1}) = f (x\textsubscript{2})\\

Then, x\textsubscript{1}\textsuperscript{2}= x\textsubscript{2}\textsuperscript{2}\\

So,x\textsubscript{1 }= x\textsubscript{2 }(as, x\textsubscript{1 }+ x\textsubscript{1 }\text{cannot be } 0) 

f(x\textsubscript{1 }) = f(x\textsubscript{2 }), hence, f(x) is one-one.

However, ‘f’ is not onto, as for 1 \in N, therefore, there is no existence of x in N : f(x) = 2x + 1.

(ii) Let f: R \rightarrow [0, \infty ), is a mapping that is defined by f(x) = \vert x \vert \\

Then, we can conclude that f(x) is not a one-one as f(2) and f(-2) are the same here.

But  \vert x \vert \geq 0, so the range is [0, \infty ].\\

Therefore, f(x) is onto.

(iii) Assume, f: R \rightarrow R , be a mapping which is defined by f(x) = x\textsuperscript{2}\\

Then we can say that f(x) is not one-one as f(1) and f (-1) are the same.

The range of f(x) is [0, \infty ).

Therefore, f (x) is neither one-one nor onto.
 

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infoexpert22

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