Short Answer TypeGiven 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.
A is the set of points in a plane.
R = {(P. Q) : distance of the point P from the origin is same as the distance of the point
Q from the origin}
= {(P, Q) : | OP | = | OQ | where O is origin}
Since | OP | = | OP |, (P, P) ∈ R ∀ P ∈ A.
∴ R is reflexive.
Also (P. Q) ∈ R
⇒ | OP | = | OQ |
⇒ | OQ | = | OP |
⇒ (Q.P) ∈ R ⇒ R is symmetric.
Next let (P, Q) ∈ R and (Q, T) ∈ R ⇒ | OP | = | OQ | and | OQ | = | OT |
⇒ | OP | = | OT |
⇒ (P,T) ∈ R
∴ R is transitive.
∴ R is an equivalence relation.
Set of points related to P ≠ O
= {Q ∈ A : (Q,P) ∈ R} = {Q ∈ A : | OQ | = | OP |}
= {Q ∈ A :Q lies on a circle through P with centre O}.
Long Answer TypeShow that the relation R in the set A = { 1, 2, 3, 4, 5 } given by
R = { (a, b) : | a – b | is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.
Short Answer TypeFor 
the set of relational numbers, define 
if and only, if a d = b c. Show that R is an equivalence relation on Q.
Long Answer Type
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