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Question
Is it true that every relation which is symmetric and transitive is also reflexive? Give reasons.
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Solution
No, it is not true.
Consider a set A = {1, 2, 3, 4} and define a relation R on A.
Symmetric relation:
R = {(1, 2), (2, 1)} is symmetric on set A.
Transitive relation:
R = {(1, 2), (2, 1), (1, 1)} is the simplest transitive relation on set A.
R = {(1, 2), (2, 1), (1, 1)} is symmetric as well as transitive relation.
But R is not reflexive here.
If only (2, 2) ∈ R, had it been reflexive.
Thus, it is not true that every relation which is symmetric and transitive is also reflexive.
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