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Functions Revisited - Functions and Relations - Set Viewpoint explainedThe question of when does a table, an arrow diagram, a set of order of ordered pairs, a curve or set in the plane, describe how one the value of a first number determine that the value of a second, sets conditions on the table, arrow diagram, set of order pairs, or set of points in the plane. For a set of points in a rectangular coordinate system, the condition is called the vertical line rule. For sets the vertical line rule provides a way or method to read the value of the dependent variable from that of the independent variable. When the vertical line rule or equivalent conditions are satisfied, the table, an arrow diagram, a set of order of ordered pairs, a curve or set in the plane may be read (or used) to find the value of the second number from the first the first, when the first is given. Thus a function or computation rule is given. There may be more than one way, so called equivalent ways, to describe the same computation rule. Computation rules may also be given by formulas. When a table, an arrow diagram, a set of order of ordered pairs, a curve or set in plane satisfies the vertical line rule or equivalent condition, or not, the same table, arrow diagram, set of order of ordered pairs, a curve or set in plane may allow the value of a first number to limit or restrict the possible values of the second number. That establishes a connection, link or relation between the values of the first and second numbers. Relations may also be given or implied by equations. Now a table, arrow diagram, set or curve in the plane, the solution set of equations in two real variables can be all be represented by a set S of ordered pairs. The set will then represent or codify a function or relation as a set of ordered pairs. That is the case too when the value of a second variable y is is given formula y = f(x) in terms of a so-called first variable x. The equation y = f(x) determines a set of ordered pairs (x,y) in the plane. A modification of the vertical line method for computing functions from their codification as a set of ordered pairs. may also be used to indicate the limits or restrictions given by a relation. Following set theory, we say the set S of ordered pairs is a function (that is, gives a computation or correspondence rule) when the vertical line condition (or its set-expressed equivalent) holds. In the latter case, a vertical line method can be use to read the set S to determine the value of a second variable y from the value of the first variable x. So we have a computation rule. On the other hand, when the vertical rule fails, the vertical line method for computing y from x fails too, but it still can be used to describe the limits or restriction imposed on the value of y from knowledge of the value of the first variable x. The language of sets appear in a ritual or rote manner in modern mathematics curriculums for high school and college. The foregoing exposition is intended to provide a context for the sudden high school or college mathematics definition or perspective of a relation being a set of ordered pairs and function as special kind of relation, a set of ordered pairs with the vertical line property. The advanced study of mathematics includes and takes advantage of the set language description of ideas, there are some advantages. But identification of functions and relations with sets of ordered pairs in high school mathematics is a bit too sudden. The foregoing provides a context for it - the first draft of an improvement in the exposition of high school mathematics. |
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