Give an example of a function fromNtoNthat isa)one-to-one but not onto. b)onto but not one-to-one. c)both onto and one-to-one (but different from the iden-tity function).d)neither one-to-one nor onto.

Step-by-step explanation:

Let f be a function from N to N.

N_set of all natural numbers

i) one to one but not onto

consider the function

When two numbers have same square we find that the numbers should be the same because they are positive.

So one to one but not onto because consider 3 it does not have square root in N.

ii) Onto but not one to one

Consider

this is onto because every number has a preimage in N.

But not onto because consider 6 and 3, f(6) = 3 and f(3) =3

So not one to one

iii) both onto and one-to-one

f(x) =

=x+1, x even

This is both one to one and onto since we consider only integers  

iv) Neither one to one nor onto

Consider the function

f(x) = 2

This is not onto because 3 cannot have a preimage in N, not one to one because f(1) = f(2) where 1 not equals 2