A={1,2,3}, find ordered pairs on A which satisfy
1) R1 is transitive
2) R2 is non-symmetric and non-antisymmetric
At first, I thought 1) is a simple question, and the result is {<1,2>,<2,3>,<1,3>}.
However, the answer tells I am wrong.
As for 2), I am not sure if or not the ordered pairs exist on A?
Any help will be greatly appreciated!
[Update]
Really thanks for Scott's help.
1) The answer for the first question is {<1,2>,<2,1>,<1,1>,<2,2>,<3,3>}
However,I don't know why the answer is above.
2)
Find a non-symmetric relation on A: {<1,2>}
Find a non-antisymmetric relation on A: {<1,3>,<3,1>}
so R2 is {<1,2>,<1,3>,<3,1>}
I am not sure if it is right, because {<1,3>,<3,1>} is symmetric.
Your first answer is not wrong: $\{\langle 1,2\rangle,\langle 2,3\rangle,\langle 1,3\rangle\}$ is a perfectly good transitive relation on $A=\{1,2,3\}$. There are many others, including all of the partial orders that you found in the other problem, not to mention the total relation $A\times A$, the set of all nine possible ordered pairs of elements of $A$.
For the second question, you need to find a relation $R_2$ on $A$ that is neither symmetric nor antisymmetric. Given two ordered pairs $\langle a,b\rangle$ and $\langle b,a\rangle$, a symmetric relation must either include both of them or neither of them. If a relation includes just one of a such a pair, it is necessarily not symmetric. Thus, you can make $R_2$ non-symmetric by including $\langle 1,2\rangle$ but not $\langle 2,1\rangle$.
A relation is antisymmetric if it always contains exactly one of two reversed pairs; if for some $a$ and $b$ it contains both $\langle a,b\rangle$ and $\langle b,a\rangle$ or neither of $\langle a,b\rangle$ and $\langle b,a\rangle$, it cannot be antisymmetric. Thus, you want to make sure that there’s some pair $a,b\in A$ such that $R_2$ contains both $\langle a,b\rangle$ and $\langle b,a\rangle$ or neither of $\langle a,b\rangle$ and $\langle b,a\rangle$. You can’t do it with $1$ and $2$, since we’ve already decided to put just one of $\langle 1,2\rangle$ and $\langle 2,1\rangle$ into $R_2$; can you do it with $1$ and $3$, say?