Question

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.

Answer 1

Answer:

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

$f(x) = x^2$

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

$f(x) = x, x odd\\f(x) = x/2, x even.$

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) = $\\\\x-1,x odd$

=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