(Quadrant II)
(Quadrant 111)
With this visual understanding, we are ready to consider the process of generation of ordered pairs. Suppose, from two given sets, x = 11,2} and v = ¡3, 4). we wish to form all the possible ordered pairs with the first element taken from set* and the second element taken from set;1. The result will, of course, be the set of four ordered pairs (1? 3)7 (1,4)„ (2, 3). and (2, 4). This set is called the Cartesian product (named after Descartes), or direct product, of the sets x and y and is denoted by x x y (read: "r crossy"). It is important to remember that, while * and v are sets of numbers, the Cartesian product turns out to be a set of ordered pairs. By enumeration, or by description, we may express this Cartesian product alternatively as
The latter expression may in fact be taken as the general definition of Cartesian product for any given sets x and y.
To broaden our horizon, now let both x and y include all the real numbers. Then the resulting Cartesian product x x y = {(a, h) | a e R and h € R\ (2.3)
will represent the set of all ordered pairs with real-valued elements. Besides, each ordered pair corresponds to a unique point in the Cartesian coordinate plane of Fig. 2.4, and, conversely, each point in the coordinate plane also corresponds to a unique ordered pair in the set x x y. In view of this double uniqueness, a one-to-one correspondence is said to exist between the set of ordered pairs in the Cartesian product (2.3) and the sel of points in the rectangular coordinate plane. The rationale for the notation x x y is now easy to perceive; wc may associate it with the crossing of the a axis and the y axis in Fig. 2.4. A simpler way of expressing the set x x y in (2.3) is to write it directly as R x R; this is also commonly denoted by R2.
Rxtending this idea, wc may also define the Cartesian product of three sets x, y, and r as follows:
which is a set of ordered triples. Furthermore, if the sets x,y? and z each consist of all the real numbers, the Cartesian product will correspond to the set of all points in a three-dimensional space. This may be denoted by R x R x R, or more simply, R-, In the present discussion, all the variables are taken to be real-valued; thus the framework will generally be R\ otR\ .,.?orr.
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