Q. 5

How many equivale

Answer :

Let A = {1, 2, 3}


A × A = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}


Now, any relation on a set A is subset of A × A.


And any equivalence relation will be reflexive, symmetric and transitive.


Now, Let B be any equivalence relation on A, containing {1, 2, 3}


Now, As B is equivalence relation on A containing (1, 2) and (2, 1), by reflexive property,


(1, 1) B


(2, 2) B


(3, 3) B


Therefore,


An equivalence relation possible is


{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}


Now, if C is another equivalence relation on A containing (1, 2) and (2, 1) then it should contain B and as well as at least one element from (1, 3), (3, 1), (2, 3), (3, 2)


Now, if C contains (1, 3) it should contain by (3, 1) by symmetry and C already contains (2, 1) therefore it should contain (2, 3) by transitive property and hence it should contain all the 4 elements.


And same case is with (3, 1), (2, 3) and (3, 2) and another possible equivalence relation is


{(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}


Hence, only two equivalence relations are possible.


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