RELATIONS

Definition 1 (Cartesian Product)

Let A & B be sets. The set A x B = {(a,b) | a element af A, b element of B} is the Cartesian product of A & B.


Definition 2 (Relation)
A relation between sets A & B is a subset R of A x B. We read aRb as "a is related to b".

Example (Equality Relation) :

The equality relation "=" defined on a set by = is the subset { (x,x) | x element of S} of S xS

Thus, for any X element of S, we have x = x

But, if x and y are different element of S, then (x,y) are not same.


Definition 3 (Partition)
A partition of a set is a collection of nonempty susets of S such that every element of S is in exactly one of the subsets.

The subsets are the cells of the partition.



Example :

Let S = {a,b,c,d,e,f,g}
Definition 4 (Equivalence Relation)

An equivalence relation R on a set S is one that satisfies these three properties for all x, y, z element of S.

1. Reflexive xRx
2. Symmetric If xRy then yRx
3. Transitive If xRy and yRx then xRz
   

sources : http://www.sandomenico.org/uploaded/photos/Library/blackboard_math.gif