The inclusion relation, denoted by ⊆, is a relation between two sets A and B where every element of A is also an element of B.

To show that the inclusion relation is a partial ordering relation on the power set of a set, we need to prove three properties:

  1. Reflexivity: Every set is a subset of itself.
  2. Antisymmetry: If A is a subset of B and B is a subset of A, then A and B are the same set.
  3. Transitivity: If A is a subset of B and B is a subset of C, then A is a subset of C.

Let P(S) denote the power set of a set S, that is, the set of all subsets of S.

  1. Reflexivity: For any set A in P(S), every element of A is also an element of A. Therefore, A ⊆ A. Hence, the inclusion relation is reflexive.
  2. Antisymmetry: Let A and B be two subsets of S such that A ⊆ B and B ⊆ A. To show that A = B, we need to show that every element of A is also an element of B, and vice versa.
  • If x is an element of A, then x is also an element of B since A ⊆ B.
  • Similarly, if y is an element of B, then y is also an element of A since B ⊆ A. Therefore, A = B. Hence, the inclusion relation is antisymmetric.
  1. Transitivity: Let A, B, and C be subsets of S such that A ⊆ B and B ⊆ C. To show that A ⊆ C, we need to show that every element of A is also an element of C.
  • If x is an element of A, then x is also an element of B since A ⊆ B.
  • Similarly, if y is an element of B, then y is also an element of C since B ⊆ C. Therefore, x is also an element of C, and hence A ⊆ C. Hence, the inclusion relation is transitive.