Is the function f(m, n) = m^2 + n^2 one-to-one?

[Cryphonus]

Hey;

im kind of stuck with the following question, i would be glad if you can help

Homework Statement

Determine if the function f : Z x Z --> Z defined by f(m, n) = m^2 + n^2 is one-to-one.

Homework Equations

if F(x1) = F(x2) implies x1=x2 then the function is one to one.

The Attempt at a Solution

i did some algebra and ended up with the equation;

(m1-m2)(m1+m2) = (n2-n1)(n2+n1)

i don't know how to disprove the equality from this point

 

[micromass]

What can you say about f(m,n) and f(n,m)?

 

[Cryphonus]

they are equal since;

m^2 +n^2 = n^2 + m^2 ?

 

[micromass]

Yes, so...

 

[Cryphonus]

so does that proves that the function is one-to-one?

 

[micromass]

No, not at all...

 

[Cryphonus]

how so?

f(m,n) = f(n,m) implies that m=n ? in this case the function should be one-to-one?

or am i missing something?

 

[HallsofIvy]

Yes, you are missing that f(m,n)= f(n,m) does NOT imply m= n.

Having shown that f(m,n)= f(n,m), what can you say about f(3,2) and f(2,3)?

And what does "one to one" mean?

 

[micromass]

For the function f to be one-to-one, you need to show that f(n,m)=f(n',m') implies n=n' and m=m'.

So, what is the negation of "f is one-to-one"?

 

[Cryphonus]

oh right, it doesn't implies that m = n;

f(3,2) is equal to f(2,3) but with different values of m and n.

By the way one to one means if whenever f (a) = f (b) then a = b? so that the every element in domain points to a single unique element in range.

 

[HallsofIvy]

Yes, exactly! Now, can you answer the question?

 

[Cryphonus]

ok since both f(3,2) and f(2,3) points to the same element in the range this function is not one-to-one. We also showed that f(m,n) = f(n,m) does not implies n = m which is basically the same thing.

i think I am right this time? ^^

Thanks a lot