If binary relations T and S are both transitive, then the binary relation $T\cup S$ is also transitive.
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My proof is as follows
This union is not transitive. So show this by a counter example.
Let T and S be relations on $N=\{1,2,3\}$.
And let us define
$(x,y)\in T $ iff $x<y$
$(x,y)\in S $ iff $x>y$
According to this definition, I construct the sets
$T=\{(1,2),(1,3),(2,3)\}$
$S=\{(2,1),(3,1),(3,2)\}$
As it is clearly seen, both T and S are transitive.
$T\cup S=\{(1,2),(1,3),(2,3),(2,1),(3,1),(3,2)\}$
$T\cup S$ is not transitive because (1,2) and (2,1) are in $T\cup S$ but (1,1) is not in $T\cup S$.
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Is this counter example true and enough to show this statement?
Also, is there any mistake in notation?
Note: I know there are many examples on this issue. But, I want to do my own example in order to understand better and to remember in the exam :)
Thank you for your great helps!