Question

in which poset P is a subset C ⊆ P such that any two elements in C are comparable?​

Answer 1

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Answer 2

Solution:

Let (P, ≤) be a poset. Elements x, y ∈ P are said to be comparable if either x ≤ y or y ≤ x.

Elements that are not comparable are said to be incomparable. A poset for which all pairs of elements are comparable is called a total order.

• Reflexivity:

Certainly, ∀A ∈ P(X), A ⊆ A.

• Transitivity:

Let A, B, C ∈ P(X). If A ⊆ B and B ⊆ C, then for all x ∈ A, we have x ∈ B,

and therefore x ∈ C. Thus A ⊆ C, as desired.

• Antisymmetry:

For A, B ∈ P(X), we have that if A ⊆ B and B ⊆ A, then A = B by double containment.