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Let $R$ be a non-empty relation on a collection of sets defined by $_{A}R_ B$ if and only if $A \cap B = \phi$. Then, (pick the true statement)

  1. $A$ is reflexive and transitive

  2. $R$ is symmetric and not transitive

  3. $R$ is an equivalence relation

  4. $R$ is not reflexive and not symmetric

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Best answer
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Let $A = \{1,2,3\}$ and $B = \{4,5\}$  and $C = \{1,6,7\}$

now $A\cap B = \emptyset$ and $B\cap C= \emptyset$ but $A\cap C\neq \emptyset$, so $R$ is not transitive.

$A\cap A = A$, so $R$ is not reflexive.

$A\cap B = B\cap A$, so $R$ is symmetric

So, $A$ is false as $R$ is not reflexive or transitive

$B$ is true.

$C$ is false because $R$ is not transitive or reflexive

$D$ is false because $R$ is symmetric
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Short trick

Empty set always symmetric relation

The correct option is B R is symmetric and not transitive
(i) Reflexive
A∩A=A≠ϕ
So; (A, A) doesn't belongs to relation R,
∴ Relation R is not reflexive.

(ii) Symmetric
If A∩B=ϕ then B∩A=ϕ is also true.
∴ Relation R is not Symmetric relation.

(iii) Transitive
If A∩B=ϕ and B∩C=ϕ, it need be true that A∩C=ϕ
For example:
A={1,2}, B={3,4}, C={1,5,6}
A∩B=ϕ and B∩C=ϕ but
A∩C={1}≠ϕ
∴ Relation R is not transitive relation.

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Answer: A

Let A = {1,2} and B = {3,4}.

Then R = {(1,3),(1,4),(2,3),(2,4)} which is not reflexive and not symmetric.
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Option B: $R$ is symmetric and not transitive


Note here, it’s written that the relation is on collection of sets, i.e. the relation is between sets not between elements of sets. So, (A,B) will be part of R iff A∩B = ∅.

So it cant be reflexive since A∩A != ∅

But it can be symmetric since A∩B = B∩A = ∅ (here A and B are disjoint sets, A={1,2} and B={3,4}

Also, its not sure that it will always be transitive, for eg. A={1,2}, B={3,4} and C={2,5}, here A∩B = ∅, B∩C = ∅ but A∩C != ∅

which makes R, symmetric but not reflexive or transitive.

Answer:

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