Graphs and Graphing

In 1637, René Descartes (1594-1650), the French mathematician and philosopher, published a book entitled Géométrie, in which he applied algebraic methods to the study of geometry. In the book, Descartes described a system (now called the rectangular coordinate system or the Cartesian coordinate system) for using points in a plane to represent ordered pairs. Given any two sets X and Y, the Cartesian product (written X × Y) of these two sets is the set of all possible ordered pairs (x,y) formed by choosing an element x from the set X and pairing it with an element y from the set Y. A relation between two sets X and Y, is a subset of their Cartesian product. To graph a relation, it is first necessary to represent the Cartesian product geometrically. Then, the graph of a particular relation is produced by highlighting that part of the representation corresponding to the points contained in the relation. Geometrically, the Cartesian product of two sets is represented by two perpendicular lines, one horizontal, one vertical, called axes. The point where the axes intersect is called the origin. Members of the set X are represented in this picture by associating each member of X with points on the horizontal axis (called the x-axis). Members of the set Y are represented by associating each member of Y with points on the vertical axis (called the y-axis). It is interesting to note that this picture is easily extended to three dimensions by considering the Cartesian product of the sets (X × Y) and Z. Z is then represented by a third axis perpendicular to the plane that represents the ordered pairs in the set (X × Y). Having established a picture of the set of all possible ordered pairs, the next step in producing a graph is to represent the subset of ordered pairs that are contained in a given relation. This can be done in a number of ways. The most common are the bar graph, the scatter graph and the line graph.