Let X be a non-empty set and P(X) be the set of all subsets of X. For A, B ∈ P(X), A R B if and only if A ∩ B=φ then the relation

Question

Let X be a non-empty set and P(X) be the set of all subsets of X. For A, B ∈ P(X), A R B if and only if A ∩ B=φ then the relation (A) R is reflexive (B) R is symmetric (C) R is transitive (D) R is an equivalence relation

Tardigrade Admin · Accepted Answer

A ∩ A=A ≠ φ if A ≠ φ, So R is not reflexive. A ∩ B=φ ⇒ B ∩ A=φ A R B ⇒ B R A ⇒ R is symmetric Note R is not transitive and not an equivalence relation.

Q. Let $X$ be a non-empty set and $P (X)$ be the set of all subsets of $X$ . For $A, B \in P (X), A RB$ if and only if $A \cap B = ϕ$ then the relation

$R$ is reflexive

$R$ is symmetric

$R$ is transitive

$R$ is an equivalence relation

$A \cap A = A \neq = ϕ$ if $A \neq = ϕ$ , So $R$ is not reflexive.
$A \cap B = ϕ \Rightarrow B \cap A = ϕ$
$A RB \Rightarrow BR A \Rightarrow R$ is symmetric
Note $R$ is not transitive and not an equivalence relation.

Q. Let X be a non-empty set and P(X) be the set of all subsets of X. For A,B∈P(X),ARB if and only if A∩B=ϕ then the relation

R is reflexive

R is symmetric

R is transitive

R is an equivalence relation