Give an example of a relation which is symmetric and transitive but not reflexive

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Solution

$L e t A = - 5, - 6 .$

$D e f i n e a r e l a t i o n R o n A a s :$

$R = (- 5, - 6), (- 6, - 5), (- 5, - 5)$

$R e l a t i o n R i s n o t r e f l e x i v e a s (- 6, - 6) \notin R .$

$R e l a t i o n R i s s y m m e t r i c a s (- 5, - 6) \in R a n d (- 6, - 5)$

$I t i s s e e n t h a t (- 5, - 6), (- 6, - 5) \in R . A l s o, (- 5, - 5) \in R .$

$∴ T h e r e l a t i o n R i s t r a n s i t i v e .$

$H e n c e, r e l a t i o n R i s s y m m e t r i c a n d t r a n s i t i v e b u t n o t r e f l e x i v e .$

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