For each of these relations on the set {21,22,23,24}, decide whether it is re - flexive, whether it is symmetric, whether it is antisymmetric, and whether it is transitive. 1. { (22, 22), (22, 23), (22, 24), (23, 22), (23, 23), (23, 24) } 2. { (21,21), (21,22), (22,21), (22,22), (23,23), (24,24) }

1. { (22, 22) (22, 23), (22, 24), (23, 22), (23, 23), (23, 24) } : Not reflective, Not symmetric, Not anti-symmetric, Transitive.

2. { (21,21), (21,22), (22,21), (22,22), (23,23), (24,24) }: Reflective, symmetric.

Explanation:

Solution

Reflective: Of every element matched to its own element

Symmetric: For every (a, b) there should be (b, a)

Anti-symmetric: For every (a, b) there should not be (b, a)

Transitive: For every (a, b) ∈R and (b, c) ∈ R - then (a, c) ER for all a, b, c ∈ A

Now,

1.{ (22, 22) (22, 23), (22, 24), (23, 22), (23, 23), (23, 24) }

Not Reflective: This is because we don't have (21,21) (23,23) and (24,24)

Not symmetric: Because we don't have (23,24) and (24,23)

Not anti symmetric: We have both (22,23) and (23,22)

Transitive: It is either 22 or 23 be (a, b) and 24 (b, a)

2. { (21,21), (21,22), (22,21), (22,22), (23,23), (24,24) }

Reflective: For all we have (a, a)

Symmetric: For every (a, b) we have (b, a)

Not Anti-symmetric

Transitive