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Checking If this Relation is Reflexive symmetric and Transitive :
Here we are going to see how to check if the given relation is reflexive, symmetric and transitive.
Reflexive, Symmetric and transitive Relation
- Is It Transitive Calculator Given a directed graph, find out if a vertex j is reachable from another vertex i for all vertex pairs (i, j) in the given graph. Here reachable mean that there is a path from vertex i to j. The reach-ability matrix is called transitive.
- What Is the Transitive Verb? A transitive verb is the general name of verbs that can answer the question of what is used when used in a sentence. If the object to which a verb is directed is open and this object can be pointed out, this verb can be described as a transitive verb. Such verbs are verbs that we use frequently in everyday life.
Let S be any non-empty set. Let R be a relation on S. Then
- R is said to be reflexive if a is related to a for all a ∈ S.
- R is said to be symmetric if a is related to b implies that b is related to a.
- 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.
Checking If this Relation is Reflexive symmetric and Transitive - Practice Questions
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”.
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Solution :
Condition for reflexive :
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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.
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