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Solution


Q.

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:               [2024]
1 symmetric and transitive only  
2 reflexive only  
3 symmetric and reflexive only  
4 symmetric only  

Ans.

(4)

   Given, A∈M⇒A∩A=ϕ, if A=ϕ

   So, R is not reflexive.

   If (A,B)∈R⇒A∩B≠ϕ⇒B∩A≠ϕ⇒(B,A)∈R

   So, R is symmetric.

   If (A,B) and (B,C)∈R⇒A∩B≠ϕ and B∩C≠ϕ

   ⇒A∩C is not necessarily non-empty set.

   Hence, A∩C not necessarily belongs to R.

   So, R is not a transitive relation.

   ∴   Given relation is symmetric and reflexive only.