relations

Question

Let R be a reflexive relation on set A

for any three elements a,b,c belongs to A

if (aRb and bRc) +> (cRa) then

which if the following is true?

a)R is symmetric but not transitive

b)R is transitive but not symmetric

c)R is an equivalence relation

d)R is neither symmetric nor transitive

Answer 1

R is a reflexive reation on set A

If system is transitive following should be true
if aRb and bRc then aRc

and it is given to us that
if aRb and bRc then cRa

hence, if system is transitive then it has to be symmetric.

If system is symmetric then if aRc then cRa has to be true
if aRb and bRc then cRa is given to us,
if system is symmetric then following has to be true
if aRb and bRc then aRc. means system has to be transitive

a & b are not possible!

(c) will be true if at least we know that either system is symmetric or transitive. but there is no such info given to us.

I think (d) will be correct answer, because we don't know if system is symmetric or transitive.

Comments

Ans C)

Given the relation is reflexive.

Now $\left ( aRb, bRb \right )\Rightarrow bRa$ // transitive

$\left ( aRb, bRc \right )\Rightarrow \left ( cRb, bRa \right )$ //Symmetric