Is this relation reflexive, symmetric, or transitive? S={(1,1),(1,2)} is a subset of Z X Z?

S is not reflexive. A relation R on Z (that is, a subset of Z x Z) is reflexive if for all n in Z it is true that (n,n) is in R. In the case of your relation S, note for example that 2 is in Z, but (2,2) is not in S. So S is not reflexive. [Note: even if you added (2,2) in, S would still not be reflexive, because e.g. 3 is in Z but (3,3) is not in S. The only way a relation on Z can be reflexive is if (n,n) is in it for _all_ integers n.]

S is not symmetric. Generally, a relation R on Z is symmetric if and only if whenever (x,y) is in R, (y,x) is also in R. Note that (1,2) is in your relation S, but (2,1) is not in S. This shows that S is not symmetric.

S is transitive. Generally, a relation R on Z is symmetric if and only if whenever (x,y) and (y,z) are in R, the pair (x,z) is also in R. Note that in the case of your relation S, the only triples (x,y,z) for which (x,y) and (y,z) are both in S are the triples (1,1,2) and (1,1,1), and in each of these cases the triple (x,z) (which is either (1,2) or (1,1), in those cases) is also in S. So S is transitive.