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  • If A = {1, 2, 3 }, define relations on A which have properties of being: (a) reflexive, transitive but not symmetric

If A = {1, 2, 3 }, define relations on A which have properties of being:

(a) reflexive, transitive but not symmetric

(b) symmetric but neither reflexive nor transitive

(c) reflexive, symmetric, and transitive.

Answers (1)

Here, A = \{ 1, 2, 3 \} .\\

(i) Assume R\textsubscript{1}= \{ (1, 1), (1, 2), (1, 3), (2, 3), (2, 2), (1, 3), (3, 3) \} \\

Here, (1, 1), (2, 2) and (3, 3)  \in \ R\textsubscript{1}. R\textsubscript{1} is reflexive.

(1, 2) \in R\textsubscript{1}, (2, 3) \in R\textsubscript{1} \Rightarrow (1, 3) \in R\textsubscript{1}

. Hence,R\textsubscript{1} is transitive.

Now, (1, 2) \in R\textsubscript{1 } \Rightarrow (2, 1) \notin R\textsubscript{1}.

Therefore, R\textsubscript{1}  is not symmetric.

(ii) Let say, R\textsubscript{2}= \{ (1, 2), (2, 1) \}  

So, (1, 2) \in R\textsubscript{2}, (2, 1) \in R\textsubscript{2}\\

Therefore, R\textsubscript{2 } is symmetric,

(1, 1) \notin R\textsubscript{2}. Therefore, R\textsubscript{2 }  is not reflexive.

(1, 2) \in R\textsubscript{2}, (2, 1) \in R\textsubscript{2 } but (1, 1) \notin R\textsubscript{2}. Hence, R\textsubscript{2 } is not transitive.

(iii) Let R\textsubscript{3 } =   \{ (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3) \} \\

R\textsubscript{3 } is reflexive as (1, 1) (2, 2) and (3, 3) \in R\textsubscript{3 }\\

R\textsubscript{3 } is symmetric as (1, 2), (1, 3), (2, 3) \in R\textsubscript{3 } \Rightarrow (2, 1), (3, 1), (3, 2) \in R\textsubscript{3 }\\

Therefore, R\textsubscript{3 }  is reflexive, symmetric and transitive.

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