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Graphing The Total ProductIn mathematics, graphing the total product enables us to understand the total product from a pictorial point of view. The Cartesian product can be used in graphing equivalence, a general property in mathematics, especially in arithmetic. The relation of equivalence has the properties: - reflexivity
- symmetry
- transitivity
The Cartesian product of a set with itself ("square product") readily exhibits the above three properties. On the other hand, the relation of total order has the properties: - irreflexivity
- asymmetry
- transitivity
- totalness
Note that the first two equivalence properties are inverted in total order; the third properties of these relations agree; and total order has an "extra" totalness property: For any two relations, one, and only one, is in total order to the other. The Cartesian graph of equivalence can be transformed into one for total order by striking out the terms which allow for reflexivity and symmetry. Given a "square product", reflexivity is graphed by the diagonal of the Cartesian product, wherein the ordered pair has the same first and second members. If the diagonal is struck out, irreflexivity results. Again, given a "square product", its diagonal separates the "square" into an upper triangular form and a lower triangular form such that upper form mirrors lower form. Thus, if (a,b) appears in upper form, then (b,a) appears in the lower form; and vice versa. Then striking out one of these forms, say, the lower triangular form, achieves asymmetry. It will then be found that the totalness property also exists: every term is either first member of an ordered pair of the form, or second member, but not both. Satisfaction of the properties of total order by such a product means that it graphs the property of total order. Result is that the upper subtriangle of a "square" Cartesian product graphs total order. (The lower subtriangle would do just as well.) TOTAL ORDERING RELATION (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) |
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