Cartesian Product and its importance in the field of Topological Robotics

In this post, I will be talking about the basics of the Cartesian Product from the viewpoint of Set Theory and how can we use its idea in Topological Robotics. So, let's start with the basic definitions of Cartesian Product and see some examples to better grasp it meaning.

Cartesian Product:

Symbolically, Cartesian Product is written as $ A \times B$. It is a set of pairs of $x$ and $y$, where $x$ belongs to set $A$ and $y$ belongs to set $B$. Mathematically, it is written as: $$ A \times B= \{(x,y) \mid x \in A, y \in B\} $$

Let's see some examples and then we can start looking at Cartesian Products in Topology.

Example 1:

Suppose we have set $A=\{ 1, 2, 8 \}$ and set $B=\{ a, b \}$. Find $A\times B$.

The answer to this problem is:

$$A\times B=\{(1,a), (1, b), (2, a), (2, b), (8, a), (8, b) \} $$

Here, we can also say that Cartesian product $A\times B$ maps all elements of $A$ to the elements of $B$. 

Example 2:

Set $A= \{ a, b, m, n \}$ and set $B= \{1 \}$. Find $B\times A$.

$$ B\times A= \{(1,a), (1, b), (1, m), (1, n) \} $$

This proves that Cartesian Product is not a commutative property when looked at in terms of Set Theory. 

Now, let's think about how Cartesian Products can be used in Topology. If we look at the Cartesian Product used in last two examples, we can see that we map elements of one set to elements of another set or we can say that we had pairs of elements. In Topology, instead of mapping elements with elements, now we will map some interval with interval or some object with other objects. Let's dive into some problems to understand it more clearly.  

Example 3:

Find the Cartesian Product $I \times I$ where $I=[0,1]$. Solve the product and show it in Cartesian Coordinates. 

The answer for this is:

$$I\times I= \{(x, y)\mid where\, x\in [0,1] \, and \, y\in [0,1] \} $$

Example 4:

Find the Cartesian Product $I \times S^1$ where $I=[0,1]$ and $S^1$ is a circle. Solve the product and show it in Cartesian Coordinates. 

$$I\times S^1= \{(x,y)\mid where \, x\in I\, and \, y\in S^1 \} $$

Example 5:

Find the Cartesian Product $S^1 \times S^1$ where $S^1$ is a circle. Solve the product and show it in Cartesian Coordinates. 

$$S^1\times S^1= \{(x,y)\mid where \, x\in S^1 and \, y\in S^1\} $$

Example 6:

Find the Cartesian Product $I \times T$ where $T$ is an alphabet. Solve the product and show it in Cartesian Coordinates. 

$$I\times T= \{(x,y)\mid where \, x\in I\, and \, y\in T \} $$

Example 7:

Find the Cartesian Product $I \times Y$ where $Y$ is an alphabet. Solve the product and show it in Cartesian Coordinates. 

$$I\times Y= \{(x,y)\mid where \, x\in I \, and \, y\in Y\} $$

Example 8:

Find the Cartesian Product $I \times M$ where $M$ is an alphabet. Solve the product and show it in Cartesian Coordinates. 

$$I\times M= \{(x,y)\mid where \, x\in I\,  and\,  y\in M\} $$

Example 9:

Find the Cartesian Product $T \times S^1$. Solve the product and show it in Cartesian Coordinates.

$$T\times S^1= \{(x,y)\mid where \, x\in T\,  and\,  y\in S^1\} $$

Example 10:

Find the Cartesian Product $S^1 \times Q$ where $Q$ is an alphabet. Solve the product and show it in Cartesian Coordinates. 

$$S^1\times Q= \{(x,y)\mid where \, x\in S^1 and\,  y\in Q\} $$

Example 11:

Find the Cartesian Product $Q \times Q$. Solve the product and show it in Cartesian Coordinates. 

$$Q\times Q= \{(x,y)\mid where \, x\in Q\, and \, y\in Q\} $$

Posted by Chintan Patel.