Consider [sets](Sets.md) $A$ and $B$ where $A$ contains elements $a$ and $B$ contains elements $b.$ ($a\in A$ and $b\in B$). The _cartesian product_ or simply the _product_ is the set containing all [ordered pairs](Ordered%20pairs.md) $(a,b)$ where $a\in A$ and $b\in B.$ This is denoted as $A\times B =\{(a,b)|a\in A, b\in B\}$^bd153e The product of a [set](Sets.md) with itself, $A\times A$ may be denoted as $A^2.$ %%Now does this affect the ordering? This is a really important question also given the examples%% # Products of finite sets If $A$ and $B$ are [finite sets](Finite%20sets.md) then $|A\times B|=|A||B|$ where $||$ denote the [their cardinality.](Cardinality.md#Finite%20set%20cardinality) # Products of multiple sets We denote the [product](Cartesian%20product.md) of multiple [sets](Sets.md) as $\prod_{i=1}^n S_i = S_1 \times S_2 \times ... \times S_n$ and the product of a set, $S,$ with itself $n$ times as $S^n.$ ^407a58 --- # Proofs and examples ## Cartesian product between $\{1,2,3\}$ and $\{3,4\}$ Given [sets](Sets.md) $A=\{1,2,3\}$ and $B=\{3,4\}$ the cartesian product is the following set of [ordered pairs](Ordered%20pairs.md) $A\times B= \{(1,3),(1,4),(2,3),(2,4),(3,3),(3,4)\}$ ## Integer coordinate plane The [cartesian product](Cartesian%20product.md) $\mathbb{Z}\times\mathbb{Z}$ forms the set of [Ordered pairs](Ordered%20pairs.md) of numbers that form $\mathbb{Z}^2.$ Such a set describes a grid with only integer coordinates. ## The Cartesian plane The [cartesian product](Cartesian%20product.md) $\mathbb{R}\times\mathbb{R}$ forms the set of [Ordered pairs](Ordered%20pairs.md) of numbers that form [$\mathbb{R}^2$](R^n#mathbb%20R%202) %%This is an iinteresting example because, I'm not sure if the pairs here are necesesarily ordered - show that they are ordered pairs. Also consider if R^n is both a set and a field or just a set.%% #MathematicalFoundations/SetTheory