1.2 The Cartesian Product

Sets

Cartesian Product

• Sets can be “multiplied” to create a new set. Better known as the cartesian product
• We can take this a step further.
  • A × B is a set of ordered pairs of elements from A and B. For example, if A = {k, m} and B = {q, r} , then
    • A × B ={ (k, q), (k, r), (l, q), (l, r), (m, q), (m, r) }
  • E.g. $\mathbb{R}$ x $\mathbb{R}$ = { (x, y) : x, y $\in$ ℝ }

Fact

• If A and B are finite sets, then |A × B| = |A| · |B|

Ordered Triple and Beyond!

• we can go even further than ordered pairs! How cool
• an ordered triple is a list.
  • E.g. $\mathbb{R}$ × $\mathbb{N}$ × $\mathbb{Z}$ = { (x, y, z) : x ∈ $\mathbb{R}$, y ∈ $\mathbb{N}$, z ∈ $\mathbb{Z}$ }
• this can be expanded to any number of sets n
  • E.g. A${}_1$ × A${}_2$ ×···× A${}_n$ = { (x${}_1$, x${}_2$,…, x${}_n$) : xi ∈ A${}_i$ for each i = 1, 2,…,n }

Space Representation

• using cartesian product, we can represent space!
• For any set A and positive integer n, the Cartesian power A${}^2$ is
  • A${}_n$ = A × A ×··· × A = { (x${}_1$, x${}_2$,…, x${}_n$) : x${}_1$, x${}_2$,…, x${}_n$ $\in$ A }
• R${}_2$ is equal to the cartesian plane and R${}_3$ is equal to three-dimensional space and so on.