This is an Abstract Algebra proofSolutionA collection S of n

This is an Abstract Algebra proof.

Solution

A collection S of nonempty subsets of a set A is a partition of A if

(1) S S ₀ = , if S and S ₀ are in S and S S ₀ , and (2) A = U {S|S S}

The equivalence classes of an equivalence relation on A form a partition of A. Conversely, given a partition on A, there is an equivalence relation with equivalence classes that are exactly the partition given. Formally the properties of an equivalence relation that motivates the definition. Such a decomposition is called a partition. For example, if we wish to identify two integers if they are either both even or both odd, then we end up with a partition of the integers into two sets, the set of even integers and the set of odd integers. To create or define an equivalence relation by merely partitioning a set into mutually exclusive subsets. The common “attribute” then might just be that elements belong to the same subset in the partition.

Let R be the relation on the set of ordered pairs of positive integers such that (a, b)R(c, d) sum of 2x and y co-ordinates gives equivalence

• R is an equivalence relation.

• The equivalence class of (x, y): [(2x, y)] = {(2xk, yk)|k Z +}.

= {(2x+r)|r R}. A r = { (x, y): y=2x+r }

• There is a natural bijection between the equivalence classes of this relation and the set of positive rational numbers.