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Transitive Relations

If A=A1,A2,A3...

Question

If $A = {A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6}}$ be the set of six unit circles with centres at $C_{1}, C_{2}, C_{3} . . . C_{6}$ as shown in the diagram. The relation $R$ on $A$ is defined by
$(A_{i}, A_{j}) \in R ⟺ C_{i} C_{j} \leq 2 \sqrt{2}, i, j \in {1, 2..., 6}$ where $C_{i} C_{j}$ represents distance between the centres $C_{i} and C_{j}$ , then

R

is symmetric and transitive but not reflexive relation.

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R

is transitive relation only.

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R

is reflexive and symmetric but not transitive relation.

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R

is neither reflexive nor transitive but it is symmetric relation

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Solution

The correct option is C $R$ is reflexive and symmetric but not transitive relation.
As distance between centres of same circle $= 0$
$\Rightarrow C_{i} C_{i} = 0 \leq 2 \sqrt{2}$
$\Rightarrow (A_{i}, A_{i}) \in R \forall i = {1, 2, 3, 4, 5, 6}$
So, $R$ is reflexive relation.

Let, $(A_{i}, A_{j}) \in R$
$\Rightarrow C_{i} C_{j} \leq 2 \sqrt{2}$
$\Rightarrow C_{j} C_{i} \leq 2 \sqrt{2}$
$\Rightarrow (A_{j}, A_{i}) \in R$
If $(A_{i}, A_{j}) \in R \Rightarrow (A_{j}, A_{i}) \in R$
So, $R$ is symmetric relation.

From diagram $(A_{1}, A_{2}) \in R$ and $(A_{2}, A_{3}) \in R$ but $(A_{1}, A_{3}) \notin R$
as, $C_{1} C_{3} = 4 > 2 \sqrt{2}$
So, $R$ is not transitive relation.

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Similar questions

Q. If

A = {A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6}}

be the set of six unit circles with centres at

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is defined by

(A_{i}, A_{j}) \in R ⟺ C_{i} C_{j} \leq 2 \sqrt{2}, i, j \in {1, 2..., 6}

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A = {A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6}}

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is-

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Transitive Relations

Standard XI Mathematics

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If A={A1,A2,A3,A4,A5,A6} be the set of six unit circles with centres at C1,C2,C3...C6 as shown in the diagram. The relation R on A is defined by (Ai,Aj)∈R⟺CiCj≤2√2,i,j∈{1,2...,6} where CiCj represents distance between the centres Ci and Cj , then