Let R be the set of real numbers.
Statement-1 : A = {(x, y) ∈ R × R : y – x is an integer} is an equivalence relation on R.
Statement-2 : B = {(x, y) ∈ R × R : x = α y for some rational number α } is an equivalence relation on R.
Text Solution
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Statement - 1 :
(i) x – x is an integer ∀ x ∈ R so A is reflexive relation.
(ii) y – x ∈ Ι ⇒ x – y ∈ Ι so A is symmetric relation.
(iii) y – x ∈ Ι and z – y ∈ Ι ⇒ y – x + z – y ∈ Ι
⇒ z – x ∈ Ι so A is transitive relation.
Therefore A is equivalence relation.
Statement - 2 :
(i) x = α x when α = 1 ⇒ B is reflexive relation
(ii) for x = 0 and y = 2, we have 0 = α for α = 0
But 2 = α (0) for no α
so B is not symmetric so not equivalence.
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