Question

Sketch the Cartesian product on the x-y plane R^2: Zx Z.

Answer 1

`\mathbb{Z}* \mathbb{Z}=\{(a,b)\lvert a, b\in \mathbb{Z}\}`

Explanation:

In general, the Cartesian product of two sets
`A,B` is a new set defined by

`A* B=\{(a,b)\lvert a\in A,b\in B\}`

The pair
`(a,b)` is ordered pair because the order is important, that is to say, in general
`(a,b)\\eq (b,a)`.

One of the most important Cartesian products in mathematics is
`\mathbb{R}* \mathbb{R}=\{(x,y) \lvert x,y \in \mathbb{R}\}` which is precisely the Cartesian Plane xy. The set
`\mathbb{Z}* \mathbb{Z}` is a subset of
`\mathbb{R}* \mathbb{R}` which is the set of all the points in the Cartesian plane whose coordinates are integers numbers. So, sketching the set
`\mathbb{Z}* \mathbb{Z}` we have a picture as the shown below.