Quiz on Equivalence Relation

Empty relations defined on a non-empty set are ____.

• not reflexive

• symmetric

• transitive

• all the above

Answer explanation

Empty relations on a non-empty set are always not reflexive, but symmetric and transitive.

Let S = {1,2,3} and ρ = {(1,1),(1,2),(2,2),(1,3),(3,1)}. Write the ordered pair(s) to be included to make S symmetric.

Answer explanation

A relation is said to symmetric if for every (a,b) in the relation must have (b,a) also. Here (1,2) is present but (2,1) is not present. So not symmetric.

Smallest equivalence relation formed using the set A={1,2,3} is AXA.

• True
• False

Answer explanation

Smallest equivalence relation formed using the set A={1,2,3} is {(1,1), (2,2), (3,3)}.
AXA is the largest equivalence relation on A.

Let S = {1,2,3} and ρ = {(1,1),(1,2),(2,2),(1,3),(3,1),(3,3), (2,1),(2,3),(3,2)}. Then ρ is ___.

• not reflexive

• not symmetric

• not transitive

• equivalence

Answer explanation

Equivalence.

• (1)

• (2)

• (3)

• (4)

Answer explanation

Which of the following is a property of an equivalence relation?

• It is reflexive

• It is antisymmetric

• It is irreflexive

• It is asymmetric

For the set B = {a,b,c}, the relation R = {(a,a),(b,b),(c,c),(a,b),(b,c)} is ___.

• an equivalence relation

• not reflexive

• not symmetric

• not transitive

Let D = {1,2,3,4} and σ = {(1,1),(2,2),(3,3),(4,4),(1,2),(2,1),(3,4),(4,3)}. Then σ is ___.

• not transitive

• not symmetric

• an equivalence relation

• not reflexive