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blocks of the partition, with the property, that every element

of A is in one of the blocks.

Using the partition P we define a binary relation on A, denoted by e(P):

Va, b ^A as(P)b <=>a and b are in the same błock of the

partition P.

The relation e(P) is reflexive, symmetric and transitive, so it is

an equivalence relation.

e(P) is called the equivalence relation associated to the

partition P.

Examples:

1.    Let A-{ 1, 2, 3, 4, 5, 6/. The partition P = fAl,A2,A3},

where: Al = (\, A}, A2 = {2, 3, 5/, A3 = {6} defines an equivalence relation s(P) on A.

For example:

\s(P)4 (because 1, 4 ^A1); 2e(F)5 (because 2, 5 ^v42); but 2 6 s(P)6 (because 2 ^A2 and 6 ^i43).

2.    Let Z - set of all integers and consider the partition

P = {A\,A2}, where: A1- set of all even integers and A2-set

of all odd integers.

P defines a relation on Z:

Vm, n £Z ms(P)n <=hn and n are both even or both odd and n have the same remainder after dividing by 2