A. Functions and Relations - Set Viewpoint

People and how they think are shaped by their language, written or verbal, and vice-versa. Set-based concepts and operations provide a language and framework for pure mathematics and some but not yet all, applications.

An  introduction or set-based view and codification of functions (how one variable may be determined by another) and of  relations (how one variables values may be related or limited by that of another) follows.  The explanations are as simple as possible but no simpler.

The explanations in the following pages not only do they introduce concepts, they also provide a  mathematical context for them. Yet the language here is artificial and technical. This set based view of mathematics shapes and provides a mould for further topics in pure and applied  mathematics.  It  has some advantages, and it required for advanced studies in mathematics.

Set Existence and Construction
(technical starting point)

A set is a collection of objects. In mathematics, the objects are required to be of a  numerical kind - constructed from numbers. Sets of objects which are not number may be used for illustration of set related ideas and concepts, but those sets are not part of pure mathematics.  In mathematics, the existence of a set  IR of real number (or sets that lead to the existence of the latter) is assumed. In pure mathematics, there are three ways to define or construct sets from existing sets.

• Set Builder, Subset Definition: If A is a set and P is condition that may be satisfied or not by elements of A, then there is proper or improper set
  
  { a in A |  condition P holds}
  
  Read the latter line as the set of p in A such that the condition P holds for a in the set A. Read the vertical bar | as such that or for which.  Variations on this notation are possible and permissible - at this level of instruction.
  
• Product Set Formation: If A and B are sets, then there exist a set of ordered pairs
  
  A x B = { (a, b) |  a in A and b in B}
  
  exists and consists of all order pairs (a,b) such that the first element (abscissa) a belongs to the first set A and the second element (ordinate) b belongs to the second set B. The case A = B leads to the set
  
  A²= A x A = { (a, b) |  both a and b are in A}
  
  For example,
  
  IR²= IR x IR = { (a, b) |  both a and b are in IR}
  
  Letters x and y may be used in place of a and b.
  
• Power Set Formation (optional): If A is a set, then there exists a set of all proper and improper subsets of A, namely
  
  P(A) =  { B |  B is a subset, proper or not, of A}

Pure mathematics essentially assumes the existence of the set of Natural numbers (Peano's axioms or axiom of infinity) and then defines or construct all further numbers using set-based definitions. That being said, secondary mathematics is and cannot be pure. The decimal view of real numbers and diagrams are needed or should be used in a mixed or applied mathematics manner to introduce and develop arithmetic, algebraic and geometric reasoning skills. The designers of the modern mathematics curricula in the 1950's onward adhered too closely to pure mathematics. 

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