Which of the following statements is true ?

Answer: The set inclusion relation ⊆ on the power set of a set S is a partial order.

• Reflexive: For any subset A of S, A ⊆ A.

• Antisymmetric: If A ⊆ B and B ⊆ A then A = B.

• Transitive: If A ⊆ B and B ⊆ C then A ⊆ C.

Why the other statements are false:

• (ℤ, ≤) is not totally ordered — This is incorrect because the usual ≤ on integers is a total order: for any integers a and b either a ≤ b or b ≤ a, and ≤ is reflexive, antisymmetric and transitive.

• (ℤ, ≠) is a poset — This is false because '≠' fails reflexivity: no element satisfies a ≠ a, so the relation cannot be a partial order.

• The directed graph is not a partial order — This claim is false. The pictured relation has self-loops (so reflexive on those vertices), no two distinct vertices related both ways (so antisymmetric), and compositions of arrows do not introduce missing relations (so transitive); therefore it represents a partial order.