First slide

Relations XII

Question

$Let A = \{A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6}\} be the set of six unit circles with centres C_{1}, C_{2}, C_{3}, \dots \dots . . C_{6}$
arranged as shown in the diagram. The relation R on A is defined by
$(A_{i}, A_{j}) \in R \Leftrightarrow C_{i} C_{j} \leq 2 \sqrt{2} then$

Moderate

A

R is symmetric, transitive but not reflexive
B

R is only transitive
C

R is symmetric, reflexive but not transitive
D

R is neither reflexive nor transitive but is symmetric

Solution

$\begin{array}{l} (i) (C_{i}, C_{i}) = 0 \leq 2 \sqrt{2}, f o r i = 1.2...6 \\ \Rightarrow (A_{i}, A_{i}) \in R \forall i R i s r e f l e x i v e \\ (i i) L e t (A_{i}, A_{j}) \in R \\ \Rightarrow C_{i} C_{j} \leq 2 \sqrt{2} \\ \Rightarrow C_{j} C_{i} \leq 2 \sqrt{2} \\ \Rightarrow (A_{j}, A_{i}) \in R \\ \Rightarrow R is symmetric \end{array}$
$(i i i) W e s e e t h a t (A_{1}, A_{2}) \in R, (A_{2}, A_{3}) \in R, b u t (A_{1}, A_{3}) \notin R . \Rightarrow R i s n o t t r a n s i t i v e$

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