cartesian product

Cartesian product

The cartesian product of two sets $A$ and $B$, denoted by $A\times B$, is the set of all possible ordered pairss $(a,b)$ where $a\in A$ and $b\in B$:
$A\times B:=\{(x,y):x\in A\wedge y\in B\}$
$B\times A:=\{(y,x):x\in A\wedge y\in B\}\ne A\times B$
e.g.: $A=\{1,2,3\}$ and $B=\{a,b\}$, then $A\times B=\{(1,a),(1,b),(2,a),(2,b),(3,a),(3,b)\}$

The cartesian product is not commutative: $A\times B\ne B\times A$

Taking the cartesian product of mutiple sets:
$A_1\times A_2\times \cdots \times A_n=\{(a_1,a_2,\dots a_n)\mid \forall i:a_i\in A_i\}$

The cardinality of the cartesian product is the product of the cardinalities of the individual sets:
$|A\times B|=|A|\cdot |B|$
So the cartesian product of a set with the empty set is the empty set: $|A\times \varnothing |=|A|\cdot 0=0⟹A\times \varnothing =\varnothing$