Visualization of Cartesian Product: $T$X$T$

Visualization  of Cartesian Product: $T$X$T$ 

By Justin Merchan

Introduction

In order to correctly understand the Cartesian product of $T$X$T$, we have to examine its elements. This Cartesian product is formed by two functions, both $T$ in this case; therefore we have to analyze the figure $T$ to determine our final Cartesian product.

Analysis of the space $T$ and $T$X$T$

The letter T contains an intersection point. In this point, all three extremes of the letter emerged. This will be important to make sure that our final product is continuous, and it is represented correctly. If we were to erase this intersection point, also called vertex (or point v) in space $T$, we would obtain three lines that follow three different directions. We think of a vertex as a point where multiple curves intersect or join together.  If we want to obtain the Cartesian product of a space product of  $T$, it should follow three different directions.

Also, the origin of the product of this two spaces should be the same. In other words, the origin of $T$X$T$ should be a space $T$ in order for the product to be continuous.

 Finally, by analyzing the dimensions where our spaces $T$ exist, we can determine the number of spaces where our Cartesian product exists. In this case, our cartesian product would exist in 4-Dimensional space.

[NEEEDS FIX] Discussion:Cartesian Product on 3-Dimensional play of a set $A$ and $B$ for a Hyperrectangle

Cartesian Product of Two Sets for a Hyperrectangle 

By Justin Merchan

Abstract 

Our objective is to create a hyperrectangle by combining two sets of ordered pairs under the concept of a Cartesian product. There will be use triplets in order to apply this practice to a three-dimensional space. 

Introduction

Given two sets $A$ and $B$, where the set of ordered pairs in $A$ are connected by a line, as well as the set of $B$. When $A \times B$, we should get an hyperrectangle (also known as a rectangular prism) Fig. 1a. 

My Solution

Given set $A=\{\alpha,\beta\}$, where $\alpha$ and $\beta$ are represented by the triplets $\alpha=(1,0,0)$ and $\beta=(1,2,0)$; times the set $B=\{a,b,c,d\}$, where $a,b,c,d$ are represented by the triplets, $a=(1,0,0)$; $b=(6,0,0)$; $c=(1,0,-4)$; and $d=(6,0,-4)$. 

Therefore, $A \times B=\{(\alpha,a);(\alpha,b);(\alpha,c);(\alpha,d);(\beta,a);(\beta,b);(\beta,c)(\beta,d)\}$   Fig. 1b.

*Please share with us your answer and final graph if possible. This is an open discussion, so if you catch a mistake, have any question regarding any of the solutions, or you have any opinion in general please comment it.