Prove that Zx is countableSolutionsimply systematic ordering

Solution

simply systematic ordering on ZxZ, then it is countable.

Let\'s just do positive integers. We can lay out all the pairs in a grid:

(1,1), (1,2), (1,3), (1,4), ...
(2,1), (2,2), (2,3), (2,4), ...
(3,1), (3,2), (3,3), (3,4), ...

Now you can draw a winding arrow through the grid to get the order:

(1,1), (1,2), (2,1), (3,1), (2,2), (1,3), (1,4), (2,3), (3,2), ...

You should be able to see the pattern as to how the arrow goes.

Well, then you\'ve put the pairs in a sequence, so they are countable.