Question

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

Answer 1

Answer:

$\mathbb{Z}\times \mathbb{Z}=\{(a,b)\lvert a, b\in \mathbb{Z}\}$

Step-by-step explanation:

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

$A\times B=\{(a,b)\lvert a\in A,b\in B\}$

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

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