Let 
be a fixed positive integer. Let a relation 
be defined in 
(the set of all integers) as follows: 
iff 
, that is, if 
is divisible by 
. Then, the relation 
is
Text Solution
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Reflexive : Since for any integer 
, we have 
is divisible by 
. Hence, 
.
is reflexive.
Symmetric : Let 
. Then, by definition of 
, where 
.
where 
.
is symmetric.
Transitive : Let 
and 
. Then, by definition of 
, we have, 
and 
, where 
.
Then it follows that
, where 
.
is transitive.
Hence, 
is an equivalence relation.
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