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Functions and Sets1 Why Set TheorySet membership, union, intersection and complements form a language for a precise description of mathematical ideas.1 Set theory is emphasized in mathematics since the axiomatic method of analysis, more precisely the arithmetic properties of integers, real numbers and functions can be logically codified and described in it. Outside calculus, you may see the role of set concepts in some presentations of analytic geometry. Lines, planes, surfaces and sometimes solid objects are regarded as set of points. Sets and set membership also have a role in combinatorics and counting, and in the combinatorial based parts of probability. Combinatorics counts or is concerned with the number of ways objects in sets can be grouped or placed together. 2 Set and Rule Viewpoints ReconciledWhen two different views of an idea are presented, they need to be reconciled. In the first instance, we could regard a function as a well-defined computational rule on a domain. The rule could be given by a formula, and the domain might be defined to the maximal set of real (or complex) numbers for which division by zero and other undefined operations are avoided. A set of ordered pairs which satisfies the vertical line property could be viewed as just another (and then a more general way) of representing or defining a function. Here
The foregoing links together and treats equally the set-theoretic definition of functions and the rule-based definition. The issue of which rule is better is not important except in the set theoretic foundations of mathematics. Both definitions should be viewed as interchangeable. 3 RelationsThe use of the word relation in set theory requires some explanation. An explanation is given here since, as suggested earlier, definitions and theories without examples to illustrate them hold vacuously. Now requiring that an equation be satisfied by (x,y) when giving the value of x restricts the value of y, and vice-versa. The satisfaction of an equation by two quantities (x,y) defines what is called a relationship between them.From equations to sets. For instance observe graphing and sketching the solutions of an equation, for example y2+2x2 = 1, defines sets of ordered pairs (a solution set) in the plane. Requiring that an equation be satisfied by the ordered pair (x,y) is equivalent to requiring that this ordered pair belongs to the solution set of the equation. In set theory, a set S of ordered pairs is said to be a relation: The knowledge that a point (x,y) belongs to a set S links (relates) the values of x and y. In particular, for a set S of points in the plane, giving the value of x restricts the value of y to the intersection of the set S with a vertical line through the point (x,0) while specifying the value of y restricts the value of x to the horizontal line through the point (0,y). This is exactly what happens when S is a solution set for an equation. Back to an Equation. Note given a set S of points, we define the characteristic function XS of the set by putting XS(x,y) = 1 whenever (x,y) belong to the set S, and putting XS(x,y) = 0 otherwise. Now the set S is the solution set of the artificial equation XS(x,y) = 1. So we can go back to the equation viewpoint of relations.
Footnotes:1But it is not
always employed in mathematics instruction. In college-level calculus texts, the
set theoretic representation of functions may be mentioned or emphasized for
function of single variable, but it is frequently omitted in the description of
functions of several variables to science and engineering student for whom
specialization in math is no longer of interest. |
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