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Q.
Total number of equivalence relations defined in the set S = {a, b, c } is :
Relations and Functions - Part 2
Solution:
The smallest equivalence relation is the identity relation $R _{1}=$
$
\{( a , a ),( b , b ),( c , c )\}
$
Then two ordered pairs of two distinct elements can be added to give three more equivalence relations.
$
R _{2}=\{( a , a ),( b , b ),( c , c ),( a , b ),( b , a )\}
$
Similarly $R _{3}$ and $R _{4}$ can be made by taking $( b , c ),( c , b )$ and (a, c), (c, a) respectively.
Finally the largest equivalence relation i.e., the universal relation
$
R _{5}=\{( a , a ),( b , b ),( c , c ),( a , b ),( b , a ),( a , c ),( c , a ),( b , c ),( c , b )\}
$
Hence, total 5 equivalence relations can be created.