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30 votes
30 votes

Let $R_1$ and $R_2$ be two equivalence relations on a set. Consider the following assertions:

  1. $R_1 \cup R_2$ is an equivalence relation
  2. $R_1 \cap R_2$ is an equivalence relation

Which of the following is correct?

  1. Both assertions are true
  2. Assertions (i) is true but assertions (ii) is not true
  3. Assertions (ii) is true but assertions (i) is not true
  4. Neither (i) nor (ii) is true

3 Answers

Best answer
33 votes
33 votes
Answer: $C$

$R_1$ intersection $R_2$ is equivalence relation..
$R_1$ union $R_2$ is not equivalence relation because transitivity needn't hold. For example, $(a, b)$ can be in $R_1$ and $(b, c)$ be in $R_2$ and $(a, c)$ not in either $R_1$ or $R_2.$
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16 votes
16 votes

$R1$ and $R2$ both are equivalence relation so $R1∩R2$ is also an equivalence relation because $\cap$ include only those pairs which are in both $R1$ and $ R2$.
Assertions (ii) is true 

$R1∪R2$ is NOT an equivalence relation 
see counter example over $(a,b)$ : 
$R1=\{(a,a),(b,b),(a,b),(b,a)\}$ is equivalence relation
$R2=\{(a,a),(b,b),(c,b),(b,c)\}$ is equivalence relation
$R1∪R2=\{(a,a),(b,b),(a,b),(b,a),(c,b),(b,c)\}$ is NOT equivalence relation because transitive pair $(a,c)$ isn't include in it .
assertions (i) is not true

Ans is C
 

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