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A function f  from the set of natural numbers to integers defined by

 is

A. onto but not one­-one

B. one­-one and onto both

C. neither one-­one nor onto

D. one­-one but not onto.

 

Answers (1)

\\\text{Given that, } f(n)=\left\{\begin{array}{l}\frac{n-1}{2}, \text { when } n \text { is odd } \\ -\frac{n}{2}, \text { when } n \text { is even }\end{array}\right. \\ \text{and }f: N \rightarrow I, \text{where N is the set of natural numbers and I is the set of integers.}

\\\text{Let x, y in N and both are even. Then, } \\ f(x)=f(y) \Rightarrow-\frac{x}{2} = -\frac{y}{2} \Rightarrow x=y \\\\ \text{Again, x, y in N and b both are odd. Then, }\\ f(x)=f(y) \Rightarrow \frac{x-1}{2}=\frac{y-1}{2} \Rightarrow x=y

\\\text{So, mapping is one-one, since, each negative integer is an image of } \\ \text{even natural number and positive integer is an image of odd natural number. } \\ \text{So, mapping is onto,Hence, mapping is one-one onto.}

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