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Let f be defined from N to N be defined by f(n) = n + 1 over 2 if n is odd and n over 2 if n is even for all n which belong to N. State whether the function f is bijective. Justify your answer.

Q.9

Let f : N \rightarrow N be defined by

                                f(n) = \left\{\begin{matrix} \frac{n+1}{2} & if\;n\;is\;odd \\ \frac{n}{2} & if\;n\;is\;even \end{matrix}\right.   for all n\in N.

State whether the function f is bijective. Justify your answer.

Answers (1)
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f : N \rightarrow N   ,  n\in N

    f(n) = \left\{\begin{matrix} \frac{n+1}{2} & if\;n\;is\;odd \\ \frac{n}{2} & if\;n\;is\;evem \end{matrix}\right.

Here we can observe,

 f(2)=\frac{2}{2}=1           and       f(1)=\frac{1+1}{2}=1

As we can see f(1)=f(2)=1  but 1\neq 2

\therefore     f is not one-one.

Let,n\in N    (N=co-domain)

case1   n be even

   For r \in N,      n=2r   

then there is 4r \in N such that f(4r)=\frac{4r}{2}=2r

case2   n be odd

For  r \in N,   n=2r+1

then there is 4r+1 \in N such that f(4r+1)=\frac{4r+1+1}{2}=2r +1

\therefore  f is onto.

f is not one-one but onto

hence, the function f is not bijective.

 

 

 

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