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.
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 TypeHere (a, b) R (c, d) ⇔ a + d = b + c.
(i) Now (a, b) R (a, b) if a + b = b + a, which is true.
∴ relation R is reflexive.
(ii) Now (a, b) R (c, d)
⇒ a + d = b + c ⇒ d + a = c + b
⇒ c + b = d + a ⇒ (c, d) R (a, b)
∴ relation R is symmetric.
(iii) Now (a, b) R (c, d) and (c, d) R (e,f)
⇒ a + d = b + c and c + f = d + e
⇒ (a + d) + (c + f) = (b + c) + (d + e) ⇒ a + f = b + e
⇒ (a , b) R (e, f)
∴ relation R is transitive.
Now R is reflexive, symmetric and transitive
∴ relation R is an equivalence relation.
For 
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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