Find inf and sup of a set with a pair of elements

Question

Find the infimum and supremum of the set D given by

$$D=\{(x,y)\in \mathbb{ZxZ}|x^2+y^2≤9\}$$

I suspect the supremum to be $9$ and the infimum to be $0$. But how do I prove it?

Answer 1

I'm not sure how you would order a set of points but I'll try to help.

The expression defining the set ($x^2+y^2\leq 9$) might look familiar. And in fact it is the Pythagorean Theorem. This means that the set can be thought as a set of right angle triangles whose hypotenuse is less than or equal to $\sqrt{9}=3$. Equivalently you can think it as the lattice points* inside a circle of radius 3.

Eitherway, you're now able to construct the set yourself. Here is the set but only including the points from the first quadrant. To get the others, you just have to add negative sign in front of the coordinates.

$$\{(0,0),(0,1),(0,2),(0,3),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2),(3,0)\}$$

For example, if you choose to order the points accroding to the sum of the coordinates, we see that the point $(2,2)$ has the greates sum, 4. In this case the point $(2,2)$ is the maximum and $(-2,-2)$ the minimum. Since the set is finite, the minimum and maximum are infimum and supremum respecticely.

* Lattice points are the points with integer coordinates.