How many equivalence relations on the set {1,2,3} containing (1,2) and (2,1) are there in all? Justify your answer.

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Filo · Accepted Answer

Total possible pairs ={(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)}Reflexive means (a,a) should be in relation.So, (1,1),(2,2),(3,3) should be in relation.Symmetric means if (a,b) is in relation, then (b,a) should be in relation.So, since(1,2) is in relation, (2,1) should also be in relationTransitive means if (a,b) is in relation and (b,c) is in relation, then (a,c) is in relation.So, if (1,2) is in relation and (2,1) is in relation, then (1,1) should be in relation.Relation R1​={(1,2),(2,1),(1,1),(2,2),(3,3)}Total possible pairs ={(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)}So, smallest relation is R1​={(1,2),(2,1),(1,1),(2,2),(3,3)}If we add (2,3)then we have to add (3,2) also, as it is symmetricbut, as (1,2) & (3,2) are there, we need to add (1,3) also, as it is transitiveAs we are adding (1,3) we should add (3,1) also, as it is symmetricRelation R2​={(1,2),(2,1),(1,1),(2,2),(3,3),(2,3),(3,2),(1,3),(3,1)}Hence, only two possible relation are there which are equivalence.

Question Text	How many equivalence relations on the set {1,2,3} containing (1,2) and (2,1) are there in all? Justify your answer.
Answer Type	Text solution:1
Upvotes	150

How many equivalence relations on the set {1,2,3} containing (1,2) and (2,1) are there in all? Justify your answer.

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