A = {2,4,5,6,9,10,12,18,30,36,60,72}
R={(a,b) | a divides b}
I want to find a total order about partially ordered set(A,R).
If there are multiple possible values, select a large number first.
In this problem, I think total order is starting with 9.
Because 9,5,2 is a lower bound, and I have to select a large number first.
Is this right? I want to check my explanation. Also, I want to know the total order about partially ordered set(A,R).
You are correct that the total order in this case should start with 9. To select the second number, observe that all elements of A that are greater than 9 are also even, so the second number can not be any of those and thus must be one of 2,4,5 or 6. Similarly, the second number in the total order can not be 4 or 6, since those must come after 2. So you are left with 2 options: 2 or 5. Since your condition forces you to pick the larger one, you must choose 5. The third number in the total order must be 2, since everything else in A is divisible by 2. To choose the fourth number, again start eliminating the bigger numbers first. To do that, note that everything that is greater than 10 in A is a multiple of 6, so it must come after 6, and therefore can not be the fourth number in the total order. This leaves only 4, 6 and 10 as the options for the fourth number. Since, there is no way to eliminate any of these, because none of them divide each other, you must pick the greatest one. After that you can keep going with the same technique and you should end up with a total order of A that satisfies your condition.