If the Given Relation is Reflexive Symmetric or Transitive :
Here we are going to see how to check if the given relation is reflexive, symmetric and transitive.
Let S be any non-empty set. Let R be a relation on S. Then
Question 1 :
Discuss the following relations for reflexivity, symmetricity and transitivity:
(iv) Let A be the set consisting of all the female members of a family. The relation R defined by “aRb if a is not a sister of b”.
Solution :
Let A be the relation consisting of 4 female members, a grand mother (a), her two children (b and c) and a grand daughter (d).
Condition for reflexive :
R is said to be reflexive, if a is related to a for a ∈ S.
a is not a sister of a itself. Hence it is reflexive.
Condition for symmetric :
R is said to be symmetric, if a is related to b implies that b is related to a.
aRb that is, a is not a sister of b.
bRa that is, b is not a sister of c.
Note : We should not take b and c, because they are sisters, they are not in the relation.
Hence it is symmetric.
Condition for transitive :
R is said to be transitive if “a is related to b and b is related to c” implies that a is related to c.
aRc that is, a is not a sister of c.
cRb that is, c is not a sister of b. But a is not a sister of b. Hence it is transitive.
Hence the given relation A is reflexive, symmetric and transitive.
(v) On the set of natural numbers the relation R defined by “xRy if x + 2y = 1”.
Solution :
Condition for reflexive :
R is said to be reflexive, if a is related to a for a ∈ S.
let x = y
x + 2x = 1
3x = 1 ==> x = 1/3
1/3 is not related to 1/3, because 1/3 is not a natural number and it is not in the relation.R is not symmetric.
The given set R is an empty relation. An empty relation can be considered as symmetric and transitive.
Hence R is not reflexive, symmetric and transitive.
Difference between reflexive and identity relation
After having gone through the stuff given above, we hope that the students would have understood, how to check whether the a relation is reflexive, symmetric or transitive"
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