Question

Define F: ℤ → ℤ by the rule F(n) = 2 − 3n, for each integer n. (i) Prove that F is one-to-one. Proof: Suppose n1 and n2 are any integers, such that F(n1) = F(n2). Substituting from the definition of F gives that 2 − 3n1 = . Solving this equation for n1 and simplifying the result gives that n1 = . Therefore, F is one-to-one.

Answer 1

Answer:

Step-by-step explanation:

(i)

To prove that  F    is   1-1 we have to show the following.

         If   F(n) = F(m)     then      n=m.   For any integers n,m.

So,  suppose that in fact

    2 - 3n  = 2 - 3m     ,  subtracting 2 on both sides  we get  

    -3n = -3m ,  dividing both sides of the inequality by 3 we get

   n = m        as we wanted.

The general idea behind a one to one function is that every image has a unique pre-image.

Let's think about a classic example of a function that is NOT one to one.

                       $f(x) = x^2$

$f$     is NOT one to one because    $4 = (-2)^2 = 2^2$

So it NOT the preimage of a single element.

That's why the condition states

If   f is one to one then

if    f(n) = f(m)      then      n=m