Let $S = {1, 2, 3, …, 10}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $R = {(A, B) : A ∩ B ≠ 𝜙; A, B ∈ M}$ is :
(A)
symmetric only
(B)
reflexive only
(C)
symmetric and reflexive only
(D)
symmetric and transitive only
MEDIUM
Correct Answer:A
Explanation
Let $S = {1, 2, 3, …, 10}$. $R = {(A, B): A∩B ≠ 𝜙; A, B∈ M}$.
For Reflexive, $M$ is subset of 'S'. So $𝜙 ∈ M$ for $𝜙 ∩ 𝜙 = 𝜙 ⇒$ but relation is $A ∩ B ≠ 𝜙$. So it is not reflexive.
For symmetric, $A R B ⇒ A ∩ B ≠ 𝜙, ⇒ B R A ⇒ B ∩ A ≠ 𝜙$, So it is symmetric.
For transitive, If $A = {(1, 2), (2, 3)}, B = {(2, 3), (3, 4)}, C = {(3, 4), (5, 6)}$. $A R B$ & $B R C$ but $A$ does not relate to $C$. So it not transitive.