Question

Which relation on the set 1, 2, 3, 4 is a partial order?

1) Reflexive relation
2) Symmetric relation
3) Transitive relation
4) Antisymmetric relation

Answer 1

The relation that is a partial order on the set {1, 2, 3, 4} is the reflexive relation.

Step-by-step explanation:

In this question, we need to determine which relation on the set {1, 2, 3, 4} is a partial order. A partial order relation must satisfy three conditions: reflexivity, antisymmetry, and transitivity. Let's consider each option:

1. The reflexive relation is a relation where every element is related to itself. In this case, the relation {(1, 1), (2, 2), (3, 3), (4, 4)} is reflexive, satisfying the first condition.
2. The symmetric relation is a relation where if (a, b) is in the relation, then (b, a) must also be in the relation. None of the options satisfy this condition.
3. The transitive relation is a relation where if (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation. None of the options satisfy this condition.
4. The antisymmetric relation is a relation where if (a, b) and (b, a) are in the relation, then a must equal b. None of the options satisfy this condition.

Therefore, the only relation that satisfies all three conditions of reflexivity, transitivity, and antisymmetry is the reflexive relation.