How to mathematically describe this weird set of points?

[Lotto]

Homework Statement

I have this set of points defined for $x , y\in \mathbb Z$. What condtitions do we need to describe it?

Relevant Equations

$|x| \le |y|$

It is clear that one part of the solution is $|x|\le |y|$, but that is not enough. We need another condition to get rid of some points. How to find it?

I tried to write down some x-values and their y-value and tried to find a pattern, but I didn't see it. Any help?

 

[FactChecker]

It looks like they are very regular. Are they multiples of some number? If so, do you know how to describe that set?

 

[Frabjous]

Try writing equations for the lines (include the bounds).

 

[fresh_42]

You can eliminate a variable by division $q:=x/y$ and write the set as $\{q\in \mathbb{Q}\,|\,|q|\leq 1\}.$ There is no pattern between numerator and denominator except $|x|\leq |y|.$ How else would you describe all rational numbers as less or equal to one? There are really, really many of them.

 

[Lotto]

I got it! Writing down equations of some lines was very helpful...

 

[FactChecker]

I got it! Writing down equations of some lines was very helpful...

What was your answer? I had to piece logic together:

Spoiler: Logic?

Let $4\mathbb{Z} = \{4i : i \in \mathbb{Z}\}$. $A=\{(x,y)\in \mathbb{Z}\times\mathbb{Z} : (|x|=|y| )\vee (( |x|\lt |y|) \wedge ((x\in 4\mathbb{Z}) \vee (y\in 4\mathbb{Z})))\}$

I think $A$ works as the answer but I would not bet the farm on it.

 

[Lotto]

What was your answer? I had to piece logic together:

Spoiler: Logic?

Let $4\mathbb{Z} = \{4i : i \in \mathbb{Z}\}$. $A=\{(x,y)\in \mathbb{Z}\times\mathbb{Z} : (|x|=|y| )\vee (( |x|\lt |y|) \wedge ((x\in 4\mathbb{Z}) \vee (y\in 4\mathbb{Z})))\}$

I think $A$ works as the answer but I would not bet the farm on it.

I have it similar, but your solution is more elegant. My is $(|x| \le |y|) \land ((|x|=|y|) \lor (x \equiv 0 \mod 4) \lor (y \equiv 0 \mod 4))$.

 

[FactChecker]

I have it similar, but your solution is more elegant. My is $(|x| \le |y|) \land ((|x|=|y|) \lor (x \equiv 0 \mod 4) \lor (y \equiv 0 \mod 4))$.

I like yours better.