Let ρ1 and ρ2 be two equivalence relations defined on a non-void set S. Then

Question

Let ρ1 and ρ2 be two equivalence relations defined on a non-void set S. Then (A) both ρ1 ∩ ρ2 and ρ1 ∪ ρ2 are equivalence relations (B) ρ1 ∩ ρ2 is equivalence relation but ρ1 ∪ ρ2 is not so. (C) ρ1 ∪ ρ2 is equivalence relation but ρ1 ∩ ρ2 is not so (D) neither ρ1 ∩ ρ2 nor ρ1 ∪ ρ2 is equivalence relation.

Tardigrade Admin · Accepted Answer

Union of two transitive may or may not be transitive

Q. Let $ρ_{1}$ and $ρ_{2}$ be two equivalence relations defined on a non-void set $S$ . Then

both $ρ_{1} \cap ρ_{2}$ and $ρ_{1} \cup ρ_{2}$ are equivalence relations

$ρ_{1} \cap ρ_{2}$ is equivalence relation but $ρ_{1} \cup ρ_{2}$ is not so.

$ρ_{1} \cup ρ_{2}$ is equivalence relation but $ρ_{1} \cap ρ_{2}$ is not so

neither $ρ_{1} \cap ρ_{2}$ nor $ρ_{1} \cup ρ_{2}$ is equivalence relation.

Union of two transitive may or may not be transitive

Q. Let ρ1​ and ρ2​ be two equivalence relations defined on a non-void set S. Then

both ρ1​∩ρ2​ and ρ1​∪ρ2​ are equivalence relations

ρ1​∩ρ2​ is equivalence relation but ρ1​∪ρ2​ is not so.

ρ1​∪ρ2​ is equivalence relation but ρ1​∩ρ2​ is not so

neither ρ1​∩ρ2​ nor ρ1​∪ρ2​ is equivalence relation.

Union of two transitive may or may not be transitive

Q. Let $ρ_{1}$ and $ρ_{2}$ be two equivalence relations defined on a non-void set $S$ . Then

both $ρ_{1} \cap ρ_{2}$ and $ρ_{1} \cup ρ_{2}$ are equivalence relations

$ρ_{1} \cap ρ_{2}$ is equivalence relation but $ρ_{1} \cup ρ_{2}$ is not so.

$ρ_{1} \cup ρ_{2}$ is equivalence relation but $ρ_{1} \cap ρ_{2}$ is not so

neither $ρ_{1} \cap ρ_{2}$ nor $ρ_{1} \cup ρ_{2}$ is equivalence relation.