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If $\mathrm{A}=\{1,2,3,4\}$, define relations on $\mathrm{A}$ which have properties of being
(i) reflexive, transitive but not symmetric.
(ii) symmetric but neither reflexive nor transitive.
(iii) reflexive, symmetric and transitive.
(i) reflexive, transitive but not symmetric.
(ii) symmetric but neither reflexive nor transitive.
(iii) reflexive, symmetric and transitive.
Solution:
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Verified Answer
Given that, $A=\{1,2,3,4\}$
(i) Let $\quad \mathrm{R}_1=\{(1,1),(1,2),(2,3),(2,2),(1,3),(3,3)\}$ $\mathrm{R}_1$ is reflexive, since, $(1,1)(2,2)(3,3)$ lie in $\mathrm{R}_1$.
Now, $(1,2) \in \mathrm{R}_1,(2,3) \in \mathrm{R}_1 \Rightarrow(1,3) \in \mathrm{R}_1$
Hence, $\mathrm{R}_1$ is also transitive but $(1,2) \in \mathrm{R}_1$ $\Rightarrow(2,1) \notin \mathrm{R}_1$.
So, it is not symmetric.
(ii) Let $\quad \mathrm{R}_2=\{(1,2),(2,1)\}$
Now, $\quad(1,2) \in \mathrm{R}_2,(2,1) \in \mathrm{R}_2$
So, it is symmetric.
(iii) Let $\mathrm{R}_3=\{(1,2),(2,1),(1,1),(2,2),(3,3),(1,3),(3,1)$, $(2,3)\}$
Hence, $R_3$ is reflexive, symmetric and transitive.
(i) Let $\quad \mathrm{R}_1=\{(1,1),(1,2),(2,3),(2,2),(1,3),(3,3)\}$ $\mathrm{R}_1$ is reflexive, since, $(1,1)(2,2)(3,3)$ lie in $\mathrm{R}_1$.
Now, $(1,2) \in \mathrm{R}_1,(2,3) \in \mathrm{R}_1 \Rightarrow(1,3) \in \mathrm{R}_1$
Hence, $\mathrm{R}_1$ is also transitive but $(1,2) \in \mathrm{R}_1$ $\Rightarrow(2,1) \notin \mathrm{R}_1$.
So, it is not symmetric.
(ii) Let $\quad \mathrm{R}_2=\{(1,2),(2,1)\}$
Now, $\quad(1,2) \in \mathrm{R}_2,(2,1) \in \mathrm{R}_2$
So, it is symmetric.
(iii) Let $\mathrm{R}_3=\{(1,2),(2,1),(1,1),(2,2),(3,3),(1,3),(3,1)$, $(2,3)\}$
Hence, $R_3$ is reflexive, symmetric and transitive.
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