UGC NET CSE | July 2018 | Part 2 | Question: 88

Question

Which of the relations on {0, 1, 2, 3} is an equivalence relation?

• A. { (0, 0) (0, 2) (2, 0) (2, 2) (2, 3) (3, 2) (3, 3) }
• B. { (0, 0) (1, 2) (2, 2) (3, 3) }
• C. { (0, 0) (0, 1) (0, 2) (1, 0) (1, 1) (1, 2) (2, 0) }
• D. { (0, 0) (0, 2) (2, 3) (1, 1) (2, 2)

Comments

In option B, instead of (1,2) there should have been (1,1) for making  it equivalence relation.

Answer 1

For a relation to be equivalent , it must be Reflexive(xRx), Symmetric(If xRy then yRx) and Transitive(If xRy and yRz then xRz) .

Option 1 is not equivalence as (1,1) is not present hence violates reflexive property. 

Option 2 is equivalence as it is reflexive,symmetric and transitive. 

Option 3 is not equivalence as it fails at reflexive property.(all the diagonal elements are not present i.e. (1,1),(2,2),(3,3),(4,4).

Option 4 is not equivalence as it fails at reflexive property.(Since (3,3) is not present)

 

 

 

 

Comments

how B transitive in nature?

Answer 2

Answer is Option B.

A is not reflexive since $(1,1)$ is not present.

C is not reflexive since $(2,2) (3,3)$ is not present. 

D is not reflexive since $(3,3)$ is not present. 

B is equivalence relation.  

 

 

 

Comments

As B does not contain (1,1) how it can be reflexive?