Question

Which relations below are asymmetric? Why?

R₁ {(1, 1), (1, 2), (2, 1), (2, 2), (3, 1), (4, 4)}
R₂ {(2, 4), (4, 2), (1,2)}
R₃ {(1, 2), (2, 3), (3, 4), (4,4)}

Answer 1

In set theory, a relation R on set A is said to be asymmetric, if no (b,a) belongs to R when (a,b) belongs to R.

As per the definition of asymmetric relation, all three relations (R1, R2, R3) are asymmetric.

Detailed Explaination

In relation R1, (3, 1) ∈ R1 but (1, 3) ∉ R1, thus not symmetric therefore R1 is a asymmetric relation

In relation R2, (1, 2) ∈ R2 but (2, 1) ∉ R2, thus not symmetric therefore R2 is a asymmetric relation

In relation R3, (1, 2) ∈ R3 but (2,1) ∉ R3 similarly, (2, 3) ∈ R3 but (3, 2) ∉ R3, (3, 4) ∈ R3 but (4,3) ∉ R3, thus not symmetric therefore R3 is a asymmetric relation