Check whether the relation R defined on the set A=\left \{ 1,2,3,4,5,6 \right \} as R=\left \{ (a,b):b=a+1 \right \} is reflexive, symmetric or transitive.

 

 

 

 
 
 
 
 

Answers (1)

A=\left \{ 1,2,3,4,5,6 \right \}\: \: \: \: \: \: \: R=\left \{ (a,b):b=a+1 \right \}

\therefore R=\left \{ (a,a+1):a,a+1\equiv \left ( 1,2,3,4,5,6 \right ) \right \}

\Rightarrow R=\left \{ (1,2),(2,3),(3,4),(4,5),(5,6) \right \}

(i) Reflexive : 

\left \{ a,a \right \}\not\equiv R\: \forall \: a

eg: \left \{ 1,1 \right \}\equiv R\: \: \: \left \{ 1,1+1 \right \}\Rightarrow \left \{ 1,2 \right \}\equiv R 

but \left \{ 1,1 \right \}\not\equiv R  Not Reflexive

(ii) Symmetric:

\left \{ a,b \right \}\equiv R\: \: \: \: and\: \: \: \left ( b,a \right )\equiv R

eg: (1,2)\equiv R\: \: but (2,1) \not\equiv R

Not Symmetric

(iii)Transitive:

(a,b)\equiv R\: \: but (b,c) \not\equiv R\: \: \: Thus

(1,2)\equiv R\: \: \: \: \left \{ 2,3 \right \}\equiv R\: \: \: \: but (1,3) \not\equiv R\: \: \: \left \{ 1,1+2 \right \}=\left \{ 1,3 \right \}\not\equiv R

Not Transitive

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