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:
Let P denote the set of all straight lines in a plane. The relation R defined by “lRm if l is perpendicular to m”.
Solution :
Condition for reflexive :
R is said to be reflexive, if a is related to a for a ∈ S.
A line will not be perpendicular to itself. Hence it is not reflexive.
Condition for symmetric :
R is said to be symmetric, if a is related to b implies that b is related to a.
lRm that is, l perpendicular to m.
mRl, m is perpendicular to l, both are true. 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.
Let l, m and n be the set of lines in P.
If “l is related to m and m is related to n” implies that l is not related to n, because they l and n are parallel lines.
So, is transitive is not true.
Hence P is relation which is reflexive but not symmetric and not transitive.
(iii) Let A be the set consisting of all the 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 elements mother (a), father (b), a son (c) and a daughter (d).
Condition for reflexive :
R is said to be reflexive, if a is related to a for a ∈ S.
Let "a" be a member of a relation A, a will be not a sister of a. Hence it is not 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.
bRc that is, b is not a sister of c.
dRc that is, d is a sister of c.
Hence it is not 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.
dRa that is, d is not a sister of a.
aRc that is, a is not a sister of c. But a is a sister of c, this is not in the relation. Hence it is not transitive.
Hence the given relation A is reflexive, but not symmetric and transitive.
Difference between reflexive and identity relation
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