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 TypeFor 
the set of relational numbers, define 
if and only, if a d = b c. Show that R is an equivalence relation on Q.
Q arbitrarily, then
(i) Since 
therefore, a, b are integers.
∴ ab = ba, since multiplication is comutative in Z.
∴ R is reflexive.
(ii) Let 
∴ a d = b c ⇒ d a = c b ⇒ c b = d a
⇒ R is symmetric.
(iii) 
∴ ad = bc and cf = de ⇒ (a d) (c f) = (b c) (d e)
⇒ (c d) (a f) = (c d) (b e), by using commutative and associative laws of multiplication in Z.
⇒ a f = be
R is transitive.
Thus R is an equivalence relation.
Long Answer Type
Switch
