Question

Prove whether f : Z × Z → Z is one-to-one AND onto if
f (m, n) = m + n + 1.
Thanks!

Answer 1

are u sure that is one-to-one ?
f(5,3)=5+3+1=9
f(4,4)=4+4+1=9
??

Answer 2

but onto is ok
because
for example f(-1,x)=-1+x+1=x
x belongs to Z so , is onto function

Answer 3

it is for sure NOT one-to-one in \(\mathbb{R}\).
Example \(f(4,6)=f(3,7)\) but \((4,6)\ne(3,7)\)

Answer 4

disregard the "in \(\mathbb{R}\)"

Answer 5

for onto

Suppose \(z\in \mathbb{Z} \), then \(z\) is either even or odd

if \(z\) is even let \(x=\frac{z}{2}\) and \(y=\frac{z}{2}-1\) then \(f(x,y)=z\)

if \(z\) is odd, let \(x=y=\frac{z-1}{2}\) then \(f(x,y)=z\).

Answer 6

oh yeah lol we had to prove whether or not it is one on one.
I proved it is onto, but not one to one