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 d c f = b e d e ⇒ (a f) (d c) = (b e) (d c)
⇒ 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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