Question

suppose f: Z x Z -> Z is defined as f(x,y)=2x+y. Is f onto? Justify.

Answer 1

I'm not to sure what the Z x Z implies, does it just mean that the domain is infinite? Does f just simplify to Z -> Z?

Answer 2

it means the first and second number \(x\) and \(y\) are both integers

Answer 3

it does not go from
\(\mathbb{Z}\to \mathbb{Z}\) because you take two integers and combine them

Answer 4

as for onto
check to see that if say \(z\) is some integer, that you can find some pair \((x,y)\) with \(2x+y=z\) it should not be that hard
one pair should spring to mind immediately

Answer 5

oh, that makes a lot more sense, so the ZxZ just means x and y have to be integers, and ZxZ->Z means I need a pair of points that also equal an integer? and it could be any pair? like (-2,10) or (0,0)?

Answer 6

* a pair of points to sub in for x and y in 2x+y

Answer 7

the easiest way would be to find a function \(g:\mathbb{Z}\to\mathbb{Z}\times\mathbb{Z}\) such that
\[ \large f\circ g=1 \]
though i haven't found one yet.

Answer 8

but I could put any set of ordered pairs that in wanted to in for x and y in 2x+y?

Answer 9

if it is not obvious that it is onto , you are thinking way too hard
give an example of some ordered pair \((x,y)\) with \(2x+y=5\) for example
just one is enough

Answer 10

i hate to disagree with @helder_edwin but you do not need to find a function going the other direction, you are not asked to prove that this is a bijection (because it isn't) just to prove than if you pick any integer \(z\), you can find a pair of integers \((x,y)\) with \(2x+y=z\)

Answer 11

let me know if this is not clear, i could write the answer and you would say "oh, is that all?"

Answer 12

No, thank you very much, I just have a bad habit of getting stuck on the small details or accidentally misinterpreting things, so I like to double check. ^_^

Answer 13

then what did you say to show it was "onto"? just curious

Answer 14

nothing, I couldn't figure out what ZxZ->Z meant, and the f(x,y) caught me off guard. I was completely hooped.
well. gotta go, almost gonna be late, thanks again!

Answer 15

yw