Let R₁ and R₂ be two equivalence relations on a set. Consider the following…
1998
Let R₁ and R₂ be two equivalence relations on a set. Consider the following assertions:
(i) R₁ ∪ R₂ is an equivalence relation.
(ii) R₁ ∩ R₂ is an equivalence relation.
Which of the following is correct?
- A.
Both assertions are true
- B.
Assertion (i) is true, but assertion (ii) is false
- C.
Assertion (ii) is true, but assertion (i) is false
- D.
Neither (i) nor (ii) is true
Attempted by 38 students.
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Correct answer: C
The intersection R₁ ∩ R₂ is always an equivalence relation. Any pair in the intersection belongs to both R₁ and R₂, so reflexivity, symmetry, and transitivity are preserved.
The union R₁ ∪ R₂ need not be an equivalence relation because transitivity may fail. For example, on {1, 2, 3}, let R₁ correspond to the partition {{1, 2}, {3}} and R₂ correspond to the partition {{1}, {2, 3}}. Then R₁ ∪ R₂ contains (1, 2) and (2, 3), but it need not contain (1, 3), so it is not transitive.
Therefore, assertion (ii) is true and assertion (i) is not necessarily true.