Consider the set is an equivalence relation Give an explicit

Consider the set

is an equivalence relation. Give an explicit description of the equivalence classes.

Solution

1.

Let, a belong to Z

a*a>0 , hence a*a belongs to N

SO, a~a

~ is reflexive

2. a~b mean ab is in N

SO, ba is in N

so, b~a

So, ~ is symmetric

3. a~b, b~c s

ab , bc is in N

HEnce, a and c have the same parity as b hence, a and c have the same partity

So, ac>0 and ac hence ac belongs to N

HEnce, a~c. So . ~ is transitive

HEnce, ~ is an equivalence relation

1. All integers which are positive belong to one equivalence class

2. All integers which are negative belong to another equivalence class