Suppose P is a partition of a set A Define a relation R on A

Suppose P is a partition of a set A. Define a relation R on A by declaring xR y if and only if x, y X for some X P. Prove R is an equivalence relation on A. Then prove that P is the set of equivalence classes of R.

Solution

Let us define the relation on the set as addition and the result being even

For the relation to be an equivalence relation, it must be reflextive, transitive and symmetric

Relation is Reflexive since (x,x) belongs to R, since (x+x) = 2x, which is divisible by 2 and hence an even number

Relation is symmetric if (y,x) belongs to R, given that (x,y) belongs to R

(x,y) belongs to R implies that x+y

(y,x) will also belong to R since x+y is same as y+x

Relation is transitive since if (a,b) belongs to R, (b,c) belongs to R, then (a,c) will also belong to R

Hence The X can be defined as the collection of some even numbers

Hence X will always be a subset of P

P is the set of equivalence classes of R since merging all the possible classes will yield the function P