The Void Relation on a Non-Empty Set A

The void relation on a non-empty set A is a subset of the Cartesian product A x A. In other words, it is a relation that contains no ordered pairs. Let's analyze the properties of the void relation.

Reflexive Property
The reflexive property states that every element in a set is related to itself. In the case of the void relation, since there are no ordered pairs, it means that no element in A is related to itself. Therefore, the void relation is not reflexive.

Symmetric Property
The symmetric property states that if (a, b) is in a relation, then (b, a) must also be in the relation. Since the void relation contains no ordered pairs, there are no elements that are related to each other. Therefore, the void relation is symmetric by vacuous truth. This means that it satisfies the symmetric property, but only because there is nothing to violate it.

Transitive Property
The transitive property states that if (a, b) and (b, c) are in a relation, then (a, c) must also be in the relation. In the case of the void relation, since there are no ordered pairs, there are no elements that are related to each other. Therefore, the void relation vacuously satisfies the transitive property.

Anti-Symmetric Property
The anti-symmetric property states that if (a, b) and (b, a) are in a relation, and a ≠ b, then the only possibility is that a and b are the same element. Again, since the void relation contains no ordered pairs, there are no elements that are related to each other, and thus the anti-symmetric property is vacuously satisfied.

Conclusion
In summary, the void relation on a non-empty set A is not reflexive, symmetric by vacuous truth, and vacuously satisfies the transitive and anti-symmetric properties. Therefore, the correct answer is option 'D' - the void relation is transitive and symmetric.