The answer is **D**.

When it comes to working with ϕ, keep it in mind that there are two possibilities:

1. The Relation R (or S in the question) is a relation on set A which itself is empty, i.e **R on A = { }**.

2. The Relation R (or S in the question) is a relation on set A which is not empty, i.e. **A have some element**s.

**Case 1: **

Here the very set on which our relation is based, A, is empty, so **there is nothing to check**, and if there is nothing to check, in mathematics and logic we take it to have all properties to be true for that. As it is, without any logical approval or explanation. And that is why we call it Vacuous Truth(wiki link), ( Vacuous means lack of intelligence! ).

So, here if any** R = { } is defined on A = { }, all the properties i.e Reflexivity, Transitivity, Symmetric, Anti-Symmetric, hold true.**

**Case 2: **

But as is asked in the question, the set** A is not empty**. So, **there are elements which need to be checked**.

So, lets take one by one...

a. **Reflexivity**: The only way Reflexivity is deemed true, is if the all the elements of must be related to each other and that is not the case here, so it is **NOT reflexive**.

b. **Transitivity and Symmetry**: These two properties are automatically implied True until and unless there is some reason for rejection. Here we have no elements at all, so there is no chance that someone can conjure up a reason to reject these. Hence, it is Symmetrical and Transitive.

Go to this link to have a detailed understanding.