6: Relations and Functions
If evolution really works, how come mothers only have two hands?
–Milton Berle
-
- 6.1: Relations
- A relation in mathematics is a symbol that can be placed between two numbers (or variables) to create a logical statement (or open sentence). The main point here is that the insertion of a relation symbol between two numbers creates a statement whose value is either true or false. For example, we have previously seen the divisibility symbol (|) and noted the common error of mistaking it for the division symbol (/).
-
- 6.2: Properties of Relations
- There are two special classes of relations that we will study in the next two sections, equivalence relations and ordering relations. The prototype for an equivalence relation is the ordinary notion of numerical equality, = . The prototypical ordering relation is ≤ . Each of these has certain salient properties that are the root causes of their importance. In this section, we will study a compendium of properties that a relation may or may not have.
-
- 6.3: Equivalence Relations
- The main idea of an equivalence relation is that it is something like equality, but not quite. Usually there is some property that we can name, so that equivalent things share that property. For example Albert Einstein and Adolf Eichmann were two entirely different human beings, if you consider all the different criteria that one can use to distinguish human beings there is little they have in common.
-
- 6.4: Ordering Relations
- The prototype for ordering relations is ≤. Although a case could be made for using < as the prototypical ordering relation. These two relations differ in one important sense: ≤ is reflexive and < is irreflexive. Various authors, having made different choices as to which of these is the more prototypical, have defined ordering relations in slightly different ways. The majority view seems to be that an ordering relation is reflexive (which means that ordering relations are modeled after ≤)
-
- 6.5: Functions
- The concept of a function is one of the most useful abstractions in mathematics. In fact, it is an abstraction that can be further abstracted! For instance, an operator is an entity which takes functions as inputs and produces functions as outputs, thus an operator is to functions as functions themselves are to numbers. There are many operators that you have certainly encountered already – just not by that name.
-
- 6.6: Special Functions
- There are a great many functions that fail the horizontal line test which we nevertheless seem to have inverse functions for. For example, x^2 fails HLT but the square root of x is a pretty reasonable inverse for it – one just needs to be careful about the “plus or minus” issue. This apparent contradiction can be resolved using the notion of restriction.