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Graphing EquivalenceGraphing equivalence is a term that describes the description of properties of mathematical concepts such as equivalence relations and total products as being analogous to the pictorial properties of graph in the Cartesian plane. We can do this because any relation is a subset of the Cartesian product of the domain and range of this relation. The definitive properties of the equivalence relation in mathematics, including in arithmetic, are: - reflexivity;
- symmetry;
- transitivity.
Instead of memorizing each property as a postulate, students can interactively learn by generatively graphing these properties using the Cartesian product, and plotting the points obtained from the Cartesian product to obtain the graph. For example, the Cartesian product of set {a,b,c} with set {1, 2} is a set of ordered pairs, each pair formed of a first member from the first set, second member from second set (Ordered pairs play the role in cartesian products that pixels do in news photographs or computer screen graphics). Thus, we calculate the cartesian product: {a,b,c} × {1,2} = {(a, 1), (a,2), (b,1), (b,2), (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, b, c} × {1, 2} | (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.
The graphing of equivalence involves the Cartesian product of a set with itself. {a, b, c, d, e} × {a, b, c, d, e} | (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 previous properties correspond, respectively, to equivalence properties: - 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).
Hence, the Cartesian product of any set with itself graphs the eqivalence relation, as students can interactively discover; that is, generating the table generates graphs of this mathematical property.
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