A function `f` from the set of natural numbers to the set of integers defined by `f(n)={(n-1)/2,\ w h e n\ n\ i s\ od d; -n/2,\ w h e n\ n\ i s\ e v e n` (a) neither one-one nor onto (b) one-one but not onto (c) onto but not one-one (d) one-one and onto both

Answer

bijective

Explanation

When n is even, let
`f(2m_(1))=f(2m_(2))`
or `-(2m_(1))/(2)=-(2m_(2))/(2)`
or `m_(1)=m_(2)`
When n is odd, let
`f(2m_(1)+1)=f(2m_(2)+1)`

or `(2m_(1)+1-1)/(2)=(2m_(2)+1-1)/(2) " or "m_(1)=m_(2)`
Therefore, `f(x)` is one-one.
Also, when n is even, `-(n)/(2)=-(2m)/(2)= -m.`
When n is odd, `(n-1)/(2)=(2m+1-1)/(2)=m.`
Hence, the range of the function is Z.
Therefore, function is onto.