Find a function satisfying the indicated properties or prove

Find a function satisfying the indicated properties or prove that no such function exists.

Solution

Let us review the definitions:

Suppose f:ABf:AB.

i) a bijective function -

f(x) = sqrt(x)

clearly this is one -one as

when sqrt(x) = sqrt(y) then x = y ,as x,y belongs to natural number

it is onto ,as for every k belong to natural there exist k² ,such that f(k²) = k

ii) Injective not surjective: x2x

We need to choose a function which is injective, so two distinct numbers will produce distinct results, and to ensure that f is not surjective we design it in a way that some number will surely not be in the range of the function.

For example: f(x)=2x would ensure that only even numbers are produced by f, so f(x)=1 is impossible. On the other hand, if f(x)=f(y) then 2x=2y, so we can divide by 2 and have x=y. Therefore f(x)=2x is injective but not surjective.

iii) Surjective not injective xx/2

We now look for a function which will produce every integer but at least two numbers will produce the same result. Such function can be dividing by 2 all the even numbers, and keeping the odd numbers in place, that is:

f(x)= {x/2 x even

{x x odd

To see that this is indeed surjective note that x=f(2x) for every xN. However this is not injective since 1=f(1)=f(2)

iv) neither injective nor surjective

f(x) = k ( constant ) clearly it is not one-one as there are many value corresponding to k,it is not onto as only 1 value k is obtained from range.