equivalence relation

Question

let P(A) be the powerset of set A.and let R be the relation on P(A) such that S₁ R S₂ when S₁ = S₂.then the relation is:

1.reflexive

2.transitive and reflexive

3.reflexive and symmetric

4.equivlence relation

here i am not getting one thing,relation is on P(A) -> P(A) or just on P(A) ? because if relation is just on P(A) then no element of P(A) will be equal to other element that is s1 != s2 and hence there will not be any element in the reltion.

I am confused with the language of the question.

Comments

let us take A = {1,2,3} thn p(a) = (∲,{1} , {2} , {3} , {1,2} , {1,3} , {2,3} , {1,2,3} )

here we cannot find any s1 which is equal to s2 but the answer of this question is a equivalence relation.

on the other hand if R is defined on p(a) * p(a) then R ⊆ P(A) * P(A)  will contain (∲ ,{{1},{1}},{{2},{2}} , ....} which will be reflexive,symm,transitive and hence equivalence relation.

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from where did u find this question...provide the refrence..

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Equality relation on set of natural numbers is an equivalence relation?

Answer 1

The power set under given relation R = {(A_i,A_j)| A_i. = A_j}. is reflexive ,symmetric,transitive... for ex A=(1,2,)  power set 2^A ={∅,(1), (2), (1,2)}

now relation will have elements = {{(∅),(∅}),  {(1),(1)}  ,{(2),(2)} ,{(1,2),(1,2)} }

and the above relation is reflexive....bcoz every element of power set is related to itself..and that wats refelxive relation demand too..and this is also symmetric ...and transitive...coz symmerty says if aRb then  bRa should also be there..it does n't says aRb should neccessarily..there inside relation.....if it belongs..then bRa...should also be there if aRb is not there inside relation then also it is symmetric.....similar for transitive... if aRb is there and bRc there..then aRc should also...be there...it does n't says aRb should neccessarily there...inside relation..in above relation..we don't have any set...like this.... so it is symmetric,transitive....and hence an equivalence relation...

Comments

thanks tauhin..i got it.and once again confirnimg,is identity relstion a equivalence relation??

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thankyou arjun sir..

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yes