Answer
-S is Reflexive
-S is not symmetric
-S is Transitive
Work Step by Step
As X be a nonempty set and P(X) the power set of X. Defining the “subset” relation S on P(X) as follows: For all A, B ∈P(X), A S B ⇔ A ⊆ B.
By telling that it is transitive,symmetric,reflexive or not.
. S is reflexive :
-S is reflexive⇔for all sub sets A of X, A S A. By definition of S, this means that for all subsets A of X, A ⊆ A. But this is true because every set is a subset of itself.
--S is not symmetric:
-S is symmetric⇔for all subsets A and B of X, if A S B then B S A. By definition of S, this means that for all subsets A and B of X, if A ⊆ B then B ⊆ A. But this is false because X (\ne=) ∅ and so there is an element a in X. As a counterexample, take A =∅, and B ={ a}.
--S is transitive:
- S is transitive⇔for all subsets A, B, and C of X, if A S B and B S C, then A S C. By definition of S, this means that for all subsets A, B, and C of X, if A ⊆ B and B ⊆ C then A ⊆ C. But this is true by the transitive property of subsets .
