The following relations are defined on the set of real numbers.
(i) 
(ii) 
(iii) 
(iv) 
(v) 
Find whether these relations are reflexive, symmetric or transitive.
Text Solution
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(i) 
is not reflexive
Symmetry 
is symmetric relation
Transitivity 
and 
Now, let a > b and b > c, then a > c
If 
and 
, then
is not transitive relation.
(ii) 
Reflexivity We have, 
is reflecxive relation.
Symmetry 
is symmetric relation.
Transitivity 
and 
is transitive relation.
(iii) 
Reflexivity For any 
, we have 
So, 
is reflexive relation.
Symmetry 
is not symmetric relation.
Transitivity 
and 
and 
is transitive relation.
(iv) 
Reflexivity Let 
is reflexive on 
.
Symmetry Let 
, then 
is symmetric on 
.
Transitivity We observe that 
and 
but 
because
is not transitive on 
.
(v) 
Reflexivity Let 
, then 
is not reflexive relation
Symmetry Now, let 
, then 
is not symmetric relation
Transitivity 
and 
and 
Then,
is transitive relation.
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