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For each relation determine if R is reflexive, symmetric, transitive, antisymmetric and if they are equivalence relations, give their equivalence classes. a) A=P(Z) and let X⊆Z be a fixed set.Define R on A by BRC if and only if B∩X=C∩X. b) Two sequences of real numbers (a_n ) and (b_n ) are eventually equal if there exists some K∈Z such that a_k=b_k for all k≥K. Let A be the set of all sequences of real numbers, and define R by (a_n ) R(b_n ) if and only if (a_n ) and (b_n ) are eventually equal.

For each relation determine if R is reflexive, symmetric, transitive, antisymmetric and if they are equivalence relations, give their equivalence classes.
 
a) A=P(Z) and let X⊆Z be a fixed set.Define R on A by BRC if and only if B∩X=C∩X.
 
b) Two sequences of real numbers (a_n ) and (b_n ) are eventually equal if there exists some K∈Z such that a_k=b_k for all k≥K. Let A be the set of all sequences of real numbers, and define R by (a_n ) R(b_n ) if and only if (a_n ) and (b_n ) are eventually equal.
 
 

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