Question

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)}

Answer 1

Answer:

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