The definitive properties of the equivalence relation in mathematics, including in arithmetic, are:

- reflexivity;
- symmetry;
- transitivity.

Thus, we calculate the cartesian product: {a,b,c} × {1,2} = {(a, 1), (a,2), (b,1), (b,3), (c,1), (c,2)}. (Cartesian product is implicit in arithmetic multiplication as shown in figurate numbers.)

The format of any Cartesian product is that of a **table**, with rows and columns, with cells at the intersection of a row and a column:

(a, 1) | (a, 2) |

(b, 1) | (b, 2) |

(c, 1) | (c, 2) |

Students can be introduced to Cartesian product by the familiar calendar:

- weeks as rows;
- weekdays as columns;
- a given day as a cell.

(a,a) | (a,b) | (a,c) | (a,d) | (a,e) |

(b,a) | (b,b) | (b,c) | (b,d) | (b,e) |

(c,a) | (c,b) | (c,c) | (c,d) | (c,e) |

(d,a) | (d,b) | (d,c) | (d,d) | (d,e) |

(e,a) | (e,b) | (e,c) | (e,d) | (e,e) |

Note the properties of this "square" table (same number of members in each set):

- The table has a
**diagonal**, containing each set element as both first and second members; - the diagonal subdivides the table into an upper subtriangular region and a lower subtriangular region;
- each element of the set appears as first member of a pair and as second member in another pair.

- The diagonal yields reflexivity (relation of element to itself);
- the relation of upper subtriangle to lower subtriangle yields symmetry;
- that each element appears as first or second element yields transitivity, as in (a,b) and (b, c) relating to (a,c).