If symmetric then transitive relation

[rajeshmarndi]

Isn't, if we have xRy and yRx then xRx will also make transitive? Because if I am right {(x,x),(y,y)} on set {x,y} is symmetric and transitive.

Isn't the above similar to, if xRy and yRz then xRz is transitive relation?

Thanks.

 

[TeethWhitener]

$R = \{(x,x),(y,y),(z,z),(x,y),(y,x),(y,z),(z,y)\}$
is an example of a relation that's symmetric and reflexive without being transitive, because
$xRy \leftrightarrow yRx$
and
$yRz \leftrightarrow zRy$
but we also have $xRy \wedge yRz$ without $xRz$.

 

[FactChecker]

Similar but not nearly as strong. xRy and yRx => xRx is a statement about a much smaller set of x and y than the transitive property requires.

 

[vivek.das2k9]

Similar but not nearly as strong. xRy and yRx => xRx is a statement about a much smaller set of x and y than the transitive property requires.

Does this mean transitive relation require atleast 3 distinct element of a set e.g {x,y,z}.

Also as I mentioned, is {(x,x),(y,y)} on set {x,y} reflexive along with symmetric and transitive.

 

[FactChecker]

(xRy and yRx) => xRx only makes a statement about the x & y where both xRy and yRx. There might easily be none of those, so it might say nothing.

PS. Even if the relation R is transitive, there may be no x & y where (xRy and yRx). An example is the order relation '>'. It's not possible for (x > y & y > x), even though '>' is a transitive relation.