Exercise 128 Here are two important denitions related to a f

Exercise 1.2.8. Here are two important denitions related to a function f : A B. The function f
is one-to-one (1–1) if a1 I= a2 in A implies that f (a1) I= f (a2) in B. The function f is onto if,
given any b B, it is possible to nd an element a A for which f (a) = b.
Give an example of each or state that the request is impossible:

(a) f : N N that is 1–1 but not onto.

(b) f : N N that is onto but not 1–1.

(c) f : N Z that is 1–1 and onto.

Solution

a)

f(n)=n+1 is 1-1 but not onto

It is not onto because there no natural number n so that

f(n)=n+1=1

b)

f(1)=1

f(2)=1

f(n)=n-1 for n>2

This is onto but clearly not 1-1 as f(1)=f(2)=1

c)

f(n)=(n-1)/2 for n odd

f(n)=-n/2 for n even

Check for onto

Case 1. z<0

f(-2z)=z

Case 2.

z>=0

f(2z+1)=z

Hence, f is onto