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Question

If A and B are two equivalence relations defined on the set C, then


A

A intersection B is an equivalence relation

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B

A intersection B is not an equivalence relation

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C

AUB is an equivalence relation

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D

AUB is not an equivalence relation

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Solution

The correct option is A

A intersection B is an equivalence relation


Explanation for the correct option :

Step 1. Equivalence relation

A relation A on a set C is called an equivalence relation if it satisfies following three properties:

  1. Relation A is Reflexive, i.e., a,aA
  2. Relation A is Symmetric, i.e., if a,bA b,aA
  3. Relation A is transitive, i.e., if a,bA and b,cA a,cA

Step 2. Example

Example: A = 1,2,3
R1 = (1,1),(2,2),(3,3),(1,2),(2,1)
R2 = (1,1),(2,2),(3,3),(2,3),(3,2)
R1∩R2 = (1,1),(2,2),(3,3)

Therefore, if R1 and R2 are equivalence relation then R1∩R2 is also an equivalence relation.

and if R1 and R2 are equivalence relation then R1∪R2 may or may not be an equivalence relation.

Hence, option "A" is correct.


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