Question:
A={1,2,3}
1) How many partial order relations can be induced over A ?
2) How many total order relations can be induced over A ?
3) Does A exist a transitive relation?
I guess total order relations over A is P(3,3)=3!=6, but I don't know how to count partial order relations over A.
Any help will be greatly appreciated!
[UPDATE]
According to Scott's reply, I get the following result,
Type 1: 3!=6
Type 2: 3
Type 3: 3
Type 4: 3
Type 5: 3!=6
So there are 21(6+3+3+3+6) partial orders on A.
Is it right? I hope someone can check it.thanks.
[UPDATE2]
Thanks for Henry's help.
Type 4:6
Type 5:1
So there are 19(6+3+3+6+1) partial orders on A.
I am not sure I am right but I hope to get corrected.
3)Does A exist a transitive relation?
I saw the answer(odd numbered problems have answers) is {<1,2>,<2,1>,<1,1>,<2,2>,<3,3>}.
However, I don't understand why <3,3> is included in the relation.