Short Answer Type(i) Since (x,y) R (x,y), ∀ (x, y) ∈ A, as x y = y x.
R is reflexive.
(ii) Again (x,y)R (u,v)
⇒ x v = y u ⇒ u y = v x and so (u, v) R (x, y).
∴ R is symmetric.
(iii) Again (x, y) R (u, v) and (u, v) R (a, b)
∴ (x, y)R (a,b).
∴ R is transitive.
From (i), (ii), (iii), it follows that R is an equivalence relation.
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.
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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