Let X be a non-empty set and let P be the collection of all subsets of X. Let R be a relation in P, defined by R = {(A, B): A is subset of B}. Then, R is
A. Reflexive relation
B. Symmetric relation
C. Not a transitive relation
D. Transitive relation
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → A and D only
Every set is a subset of itself, and if A ⊆ B and B ⊆ C, then A ⊆ C. But the empty set ∅ is a subset of the non-empty set X, while X is not a subset of ∅.
So R is reflexive and transitive, but not symmetric.
Checking the statements
(A) Reflexive relation — Correct. A ⊆ A for every A in P.
(B) Symmetric relation — Incorrect. ∅ ⊆ X, but X is not a subset of ∅.
(C) Not a transitive relation — Incorrect. A ⊆ B and B ⊆ C give A ⊆ C, so R is transitive.
(D) Transitive relation — Correct. A ⊆ B and B ⊆ C give A ⊆ C.