Here you will learn what is reflexive relation on set with definition and examples.
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What is Reflexive Relation ?
Definition : A relation R on a set A is said to be reflexive if every element of A is related to itself.
Thus, R is reflexive \(\iff\) (a, a) \(\in\) R for all a \(\in\) A.
A relation R on a set A is not reflexive if there exists an element a \(\in\) A such that (a, a) \(\notin\) R.
Note : The identity relation on a non-void set A is always a reflexive relation on A. However, a reflexive relation on A is not necessarily the identity relation on A. For example, the relation R = {(a, a), (b, b), (c, c), (a, b) is a reflexive relation on set A = {a, b, c} but it is not the identity relation on A.
Note : The universal relation on a non-void set A is reflexive.
Also Read : Identity Relation with Examples
Given below are some reflexive relation examples.
Exampleย : Let A = {1, 2, 3} be a set. Then R = {(1, 1), (2, 2), (3, 3), (1, 3), (2, 1)} is a reflexive relation on A.
But, \(R_1\) = {(1, 1), (3, 3), (2, 1), (3, 2)} is not a reflexive relation on A, because 2 \(\in\) Aย but (2, 2) \(\notin\) \(R_1\).
Example : Let X be a non-void set and P(X) be the power set of X. A relation R on P(X) defined by (A, B) \(\in\) R \(\iff\) A \(\subseteq\) B is a reflexive relation since every set is subset of itself.
Example ย : Let L be the set of all lines in a plane. Then relation R onย L defined by \((l_1, l_2)\) \(\in\) R \(\iff\) \(l_1\) is parallel to \(l_2\) is reflexive, since every line is parallel to itself.
