Topic 10 - Sets, Relations and Groups

These are more of a list of definithions than proper notes, but it's much easier to expand on an incomplete article than to start one from scratch.

A relation is an equivalence relation if it is:

Reflexive - when aRa is true for the realation

Symetric - when aRb implies bRa. For example: as a+b=c imlies that b+a is also equal to c additionis a symmetric relation.

Transitive - when aRb, bRc implies that aRc

Equivalence classes are the sets containing all elements that will produce the same result.

the order of a relation on a set S = n(S)!

function - a relation from a set A to set B where there is only on value in B that corresponds to a value in A

co-domain - all possible results of a funtion

range - the set of possible results of a function on a certain set (S). In set notation: <math> \{ y|y=f(x),x \in S \} </math>

injection - a function such that each element in the range has only one in the domain, a one-to-one mapping

surjection - a function where the co-domain is identical to the range