Cartesian Product

Definition

The cartesian product of two sets X,Y\mathbb{X}, \mathbb{Y} is an ordered pair (x,y)(x,y) where x∈Xx \in \mathbb{X} and y∈Yy \in \mathbb{Y}

Formally:

X×Y={(x,y):x∈X and y∈Y}\mathbb{X} \times \mathbb{Y} = \{ (x,y): x \in \mathbb{X} \text{ and } y \in \mathbb{Y}\}

Note: (2,5)≠(5,2)(2,5) \neq (5,2)

Example

Let X={1,2}\mathbb{X}={1,2} and Y={a,b,c}\mathbb{Y}={a,b,c}

Then:

X×Y={(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)}\mathbb{X} \times \mathbb{Y} = \{(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)\}

and:

Y×X={(a,1),(a,2),(b,1),(b,2),(c,1),(c,2)}\mathbb{Y} \times \mathbb{X} = \{(a,1),(a,2),(b,1),(b,2),(c,1),(c,2)\}

In general, if

X≠Y⇒X×Y≠Y×X\mathbb{X} \neq \mathbb{Y} \Rightarrow \mathbb{X} \times \mathbb{Y} \neq \mathbb{Y} \times \mathbb{X}

So What?

If we take the cartesian product of the Real numbers with itself we get R2\mathbb{R}^2 or the 2 dimensional real plane. Intuitively, higher dimensional analogs of sets can be constructed using solely the base set as well as this Cartesian Product operation.

Formally:

R×R=R2\mathbb{R} \times \mathbb{R} = \mathbb{R}^2