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Transitive Relations

A relation R ...

Question

A relation R is said to be circular if aRb and bRc together imply cRa.
Which of the following options is/are correct?

If a relation S is reflexive and symmetric, then S is an equivalence relation.

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If a relation S is transitive and circular, then S is an equivalence relation.

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If a relation S is circular and symmetric, then S is an equivalence relation.

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If a relation S is reflexive and circular, then S is an equivalence relation.

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Solution

The correct option is D If a relation S is reflexive and circular, then S is an equivalence relation.
Let S be reflexive and circular. Let us checking symmetry:

$S y m m e t r y :$

$L e t x S y$
$N o w s i n c e S i s r e f l e x i v e y S y t r u e .$
$S o x S y a n d y S y i s t r u e$
$N o w b y c i r c u l a r p r o p e r t y w e g e t, y S x$
$S o x S y \Rightarrow y S x$
$S o S i s s y m m e t r i c$

$T r a n s i t i v e :$

$L e t x S y a n d y S z$
$N o w b y c i r c u l a r p r o p e r t y w e g e t z S x$
$a n d b y s y m m e t r y p r o p e r t y p r o v e d a b o v e,$
$w e g e t$
$z S x \Rightarrow x S z$
$S o x S y a n d y S z \Rightarrow x S z$
$S o S i s t r a n s i t i v e .$
$S o S i s r e f l e x i v e, s y m m e t r i c a n d$
$t r a n s i t i v e a n d h e n c e a n e q u i v a l e n c e r e l a t i o n .$
$S o o p t i o n (a) i s t r u e .$

$O p t i o n (b) : L e t S b e c i r c u l a r a n d s y m m e t r i c .$
$L e t S b e d e f i n e d o n s e t {1, 2, 3}$
$N o w e m p t y r e l a t i o n i s c i r c u l a r a n d s y m m e t r i c b u t n o t r e f l e x i v e .$
$S o S n e e d n o t b e a n e q u i v a l e n c e r e l a t i o n .$
$S o o p t i o n (b) i s f a l s e .$

$O p t i o n (c) : L e t S b e t r a n s i t i v e a n d c i r c u l a r .$
$L e t S b e d e f i n e d o n t h e s e t {1, 2, 3}$ .
$N o w e m p t y r e l a t i o n a g a i n s a t i s f i e s t r a n s i t i v e a n d c i r c u l a r b u t i s n o t r e f l e x i v e .$
$S o S n e e d n o t b e a n e q u i v a l e n c e r e l a t i o n .$
$S o o p t i o n (c) i s f a l s e .$

$O p t i o n (d) : R e f l e x i v e a n d s y m m e t r i c n e e d n o t b e t r a n s i t i v e$
$f o r e x a m p l e o n {1, 2, 3}$
$S = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}$
$i s r e f l e x i v e a n d s y m m e t r i c .$
$B u t i t i s n o t t r a n s i t i v e b e c a u s e (1, 2) a n d (2, 3)$
$b e l o n g t o S b u t (1, 3) d o e s n o t .$
$S o o p t i o n (d) i s f a l s e .$

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