
1. ALGEBRA
Def A relation between a set S and a set T is a subset R of ST; and for (s,t) in R we write s R t. Thus R={(s,t)s R t}.
A relation R on SS (sometimes called simply a relation on S) is:
1.2.1. Definition of partitions
Def A partition on S is a collection of disjoint subsets of S whose set union is S, i.e. such that and
The partition is the measure of information.
We refer to the sets of as blocks of and designate the block which contains s by .
1.2.3. Operations with partitions
If s and t are in the same block of , we write:
The computation of
The computation of : to
compute we proceed inductively.
Let
and for i>1 let
.
Then
for any
i such that .
and
For and on S we say that is larger than or equal to and write if and only if every block of is contained in a block of . Thus if and only if and if and only if .
Operations "·" , "+" and the ordering of partitions form the basic link between machine concepts and algebra.
Examples of operationsLast update: 3 August, 2004