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  • Given a non empty set X, consider P(X) which is the set of all subsets of X. Define the relation R in P(X) as follows: For subsets A, B in P(X), ARB if and only if A ⊂ B. Is R an equivalence relation on P(X)? Justify your answer.

Q.8 Given a non empty set X, consider P(X) which is the set of all subsets of X. Define the relation R in P(X) as follows:
For subsets A, B in P(X), ARB if and only if A \subset B. Is R an equivalence relation
on P(X)? Justify your answer.

Answers (1)

best_answer

Given a non empty set X, consider P(X) which is the set of all subsets of X.

Since, every set is subset of itself , ARA  for all A \in P(x)

\therefore  R is reflexive.

Let ARB \Rightarrow A\subset B

This is not same as B\subset A

If  A =\left \{ 0,1 \right \}      and    B =\left \{ 0,1,2 \right \}

then we cannot say that B is related to A.

\therefore R is not symmetric.

If ARB \, \, \, and \, \, \, BRC, \, \, then \, \, A\subset B \, \, \, and \, \, B\subset C

this implies A\subset C  = ARC

\therefore R is transitive.

Thus, R is not an equivalence relation because it is not symmetric.

 

 

 

 

 

 

 

 

 

 

 

 

 

Posted by

seema garhwal

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