How to solve Relations / Reflexsive problem?

Question

Define the following relation on the set of positive integers

 

(x,y) ε R if x - y is an even integer

 

1.  Show that R is an reflexive, symmetric and transitive.

 

This relation is an equivalence relation because it is reflexive symmetric and transitive.  With an equivalence relation, the set on which the relation is defined is divided into subsets called equivalence classes.  These subsets consist of all elements that are related to each other.  The equivalence class of 1, denoted by [1] consists of all elements that are equivalent to 1 under the relation.  

 

2. How many distinct equivalence classes are there in this example? Can you describe the sets?

Answer 1

The various sets are

• The set I of all integers, x in I, y in I (I in R, in other words real numbers include all integers as a subset)
• The set E of all even integers (y in E, x in E and x-y in E)
• The set O of all odd integers (x in O, y in O, but x-y in E)
• E + O = I (that is, E v O=I, or the union of E and O is I)

The implication is that all even integers can be considered as the difference between two even numbers or two odd integers.