Q

# 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 $n\in N$.

Views

$f : N \rightarrow N$   ,  $n\in N$

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

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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