FN: Sets - Continued

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C. Functions without and with Sets

1. Description of Assignment and Computation Rules without an emphasis on sets of ordered pairs

The modern, pure mathematics, set theoretic concepts or codification of relations and function needs to be postponed or explained until a context is provided.

Assignment and computation rules (also known as functions, mappings, matching, transformation, correspondences, input-output rules) can be given in many different ways including the following:

Arrow Diagrams

Table of Values

Function Notation

Graphically by finite sets of points plotted in the plane

Graphically by smooth curves in the plane - infinite sets of ordered pair,

Formulas - Algebraic Rules

Verbal Rules.

Indicate to students that more ways to define computation and assignment rules are possible.

2. Transition to the formal set viewpoint

All ways above can be described or codified as sets of ordered pairs, finite or not, which satisfy a vertical line rule, a rule that can be described in terms of a set property.

Suppose S is a subset of set- product

A x B = { (a, b) | a in A and b in B}

Then S satisfies the "vertical line property" when and only when for each a in A, there is at most one y in B (possibly none) for which (a,y) belongs to S.

The vertical line property or vertical line rule in principle gives a way or method to evaluate or find the assigned or computed value of a function from a knowledge of the set without using the original form, if any, of the rule.

Context for the Set-Based Representation

Pure mathematics 1903 onward has been codified in terms of sets and set-based language. For better or worse, the description or codification of functions a sets which satisfy the vertical line rule or equivalent set theoretic property is part of the recent or modern codification of pure mathematics.

The representation of assignment and computation rules or functions by sets which satisfy the vertical line rule aids in defining or describing more and more functions in secondary IV and V mathematics, and in first courses in calculus. It also aids in advance courses on combinatorics - how to count - and in probability - how to calculate probabilities. Apart from the discussion of functions, the probability of events and outcomes can also be described precisely or more precisely with the aid of sets. Continue reading for more details.