# Cartesian Product

The cartesian product of two sets `A` and `B` is
the set of all ordered pairs (`a`,`b`)
with `a` and `b` elements of
`A` and `B` respectively. This new set, the cartesian product,
is usually denoted `A` × `B`. Formally:

`A` × `B` : = {(`a`,`b`) | `a` ∈ `A`, `b` ∈ `B`}.

This definition may be extended in the obvious way to form the cartesian product of more than two sets.

## ~ Group

Given two groups `G`_{1}
and `G`_{2}, with corresponding operations
+_{1} and +_{2}, we can define the cartesian product
product group `G` = `G`_{1} ×
`G`_{2}. Its elements are those of the
cartesian product of the sets of
elements of the two original groups. The group operation +, is defined
by:

(`x`_{1}, `x`_{2}) + (`y`_{1}, `y`_{2})
: = (`x`_{1} +_{1} `y`_{1} , `x`_{2} +_{2} `y`_{2}).

It can be easily varified that `G` is indeed a group.