Cartesian Products
Definition
Definition: The Cartesian Product \(A\times B\) of two sets \(A\) and \(B\) is the set of ordered pairs \((a,b)\) where \(a\in A\) and \(b\in B\). \[ A\times B =\{(a,b):a\in A, b\in B\} \]
Example
\(A=\{1,2,3\}\) and \(B=\{x,y\}\). What is \(A\times B\)?
Example
\(\mathbb{R}\times\mathbb{R}= \{(a,b): a\in \mathbb{R}, b\in\mathbb{R}\}\)
Example
What is \(\mathbb{N}\times\{-1,1\}\)?
Example
\(\mathbb{Z}\times\mathbb{Z}\)
Example
\(\mathbb{N}\times (\mathbb{N}\times\mathbb{N})\) vs \(\mathbb{N}\times\mathbb{N}\times\mathbb{N}\)
Cartesian Powers
\(A=\{H,T\}\). What is \(A^{4}\)? What is \(|A^{4}|\)?.