Here you will learn what is equivalence relation on a set with definition and examples.
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What is Equivalence Relation ?
Definition : A relation R on a set A is said to be an equivalence relation on A iff it is
(i) reflexive i.e. (a, a) \(\in\) R for all a \(\in\) A.
(ii) symmetric i.e (a, b) \(\in\) R \(\implies\) (b, a) \(\in\) R for all a, b \(\in\) A.
(iii) transitive i.e. (a, b) \(\in\) R and (b, c) \(\in\) R \(\implies\) (a, c) \(\in\) R for all a, b, c \(\in\) A.
Also Read : Types of Relations in Math
Example : If A = {1,2,3}, then the relation R = {(1,1),(2,2),(3,3),(2,1),(1,2),(2,3),(1,3).(3,2),(3,1)} is the equivalence relation on A,
because {(1,1),(2,2),(3,3)} \(\in\) R hence it is reflexive,
{(2,1),(1,2),(2,3),(3,2),(1,3),(3,1)} \(\in\) Rย hence it is symmetric on A,
{(1,2),(2,3),(1,3)} and {(1,3),(3,2),(1,2)} \(\in\) R hence it is transitive.
Example : Let R be a relation on the set of all lines in a plane defined by \((l_1, l_2)\) \(\in\) R \(\iff\) line \(l_1\) is parallel to line \(l_2\). Show that R is an equivalence relation.
Solution : Let L be the given set of all lines in a plane. Then, we observe the following properties.
Reflexive : For each line l \(\in\) L, we have
l || l \(\implies\) (l, l) \(\in\) R for all l \(\in\) L
\(\implies\) R is reflexive.
Symmetric : Let \(l_1\), \(l_2\) \(\in\) L such that \((l_1, l_2)\) \(\in\) R. Then,
\((l_1, l_2)\) \(\in\) R \(\implies\) \((l_1\) || \(l_2)\) \(\implies\) \((l_2\) || \(l_1)\) \(\in\) R.
So, R is symmetric on L.
Transitive : Let \(l_1\), \(l_2\), \(l_3\) \(\in\) L such that \((l_1, l_2)\) \(\in\) R and \((l_2, l_3)\)ย \(\in\)ย R. Then,
\((l_1, l_2)\) \(\in\) R and \((l_2, l_3)\) \(\in\) R \(\implies\) \(l_1\) || \(l_2\) and \(l_2\) || \(l_3\) \(\implies\) \(l_1\) || \(l_3\) \(\implies\) \((l_1, l_3)\) \(\in\) R.
So, R is transitive on L.
Hence, R being reflexive, symmetric and transitive is an equivalence relation on L.
