Question

Let denote the set of all natural numbers and be the relation on defined by . Check weather is an equivalence relation.

Medium

Solution

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Let R be defined on as 
.     ....(1)

Reflexivity:
We can write for all
Since, sum and product of natural numbers obeys commutative property
Hence, by def (1), we can write 
  for all
Hence, is reflexive.

Symmmetry :
Let  
   (Since, sum and product of natural numbers obeys commutative property)
or
Hence, is symmetric 

Transitivity :
Let
Let and
and
and
Hence , is transitive

is Equivalence Relation.

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