(1) (Z,≼ ) is a not totally ordered set ------> is false
actually totally ordered set means it should be a poset and each and every two elements are comparable.
http://mathworld.wolfram.com/TotallyOrderedSet.html
(2) THe set inclusion relation ⊊ is a poset on the power set of a set S. ---> True
( powerset of a set, ⊊ ) ---- satisfies reflexivity, anti-symmetric and transitivity
(3) (Z, ≠ ) is a poset ---- false
it does satisfies symmetric and does not satisfies Anti-Symmetric, counter example is, it contains (2,1) and (1,2)
(4) GIVEN DIRECTED GRAPH IS not poset -----> FALSE
it contains {(a,a),(a,b),(b,b)} --- which is reflexive, Antisymmetric and Transitive ===> it is a poset
∴ Option B is true