abstractmath.org
help with abstract math
Produced by Charles Wells. Home Website TOC Website Index Blog
Posted 30 August 2009
RELATIONS

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.
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 consciousnessraising, that are used in all the other sections.
Operations on relations Union, intersection, opposite and composition.
Properties of relations Reflexive, symmetric, transitive, antisymmetric, 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.