help with abstract math

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Posted 30 August 2009




The symbols “=” and “<” stand for relations on numbers.  The property that makes them relations is that you can make statements about two numbers using them, for example “2 < 3” and “2 = 3”.  Some of these statements are true (for example “2 < 3”) and some are false (for example “2 = 3”). 


The mathematician’s concept of relation has “<” and “=” as examples but is more abstract and general, in the same way that the mathematician’s concept of function is more general than just the functions you studied in calculus. 


Note:  In abstractmath.org I talk about only relations between two objects.  They are called binary or dyadic relations in the math literature.  See the Wikipedia article on binary relations for many more examples of and constructions on relations.

As with “=” and “<” , you use a binary relation to make a statement about two objects.  More general relations involving n objects are widely studied, particularly in computing science.  See the Wikipedia article on finitary relations for more about this.

Table of contents

Relations: Basics  The definition and basic notation used with relations.  Watch out: the definition may be surprising and difficult to understand.

Relations: Examples  This section collects many examples, some trivial, some familiar, and some consciousness-raising, that are used in all the other sections.

Operations on relations  Union, intersection, opposite and composition.

Properties of relations  Reflexive, symmetric, transitive, anti-symmetric, and others. 

Equivalence Relations  Equivalence relations, which are the same thing as partitions, are the most important type of relation in mathematics research. 

An ordering is another important type of relation that you can find out by reading the Wikipedia article on order theory.  Warning:  the word “order” has other meanings in math; see the Glossary.