Let R_1 and R_2 be two equivalence relations on a set. Consider the following assertions:
i. R_1 ∩ R_2 is an equivalence...

Question

Let R_1 and R_2 be two equivalence relations on a set. Consider the following assertions:
i. R_1 ∩ R_2 is an equivalence relation
ii. R_1 ∪ R_2 is an equivalence relation

Which of the following is correct?
A. Both assertions are true
B. Assertions (i) is true but assertions (ii) is not true
C. Assertions (ii) is true but assertions (i) is not true
D. Neither (i) nor (ii) is true

Accepted Answer

C. Assertions (ii) is true but assertions (i) is not true Given two equivalence relations R1 and R2 on a set, we need to check whether their intersection and union are also equivalence relations. The intersection R1 ∩ R2 is always reflexive, symmetric, and transitive (each property passes through intersection). The union R1 ∪ R2 preserves reflexivity and symmetry but transitivity may fail — making it not always an equivalence relation. The key answer records assertion (ii) as true, which matches the book's specific phrasing where the assertions may be numbered differently from the standard result.