Step-by-step explanation:
Consider the provided set {2,3,4,6,8,9,10,12}
Let the set is A.
![A={(2 \prec 4), (2 \prec 6), (2 \prec 8), (2 \prec 10), (2 \prec 12), (3 \prec 6), (3 \prec 9), (4 \prec 8), (4 \prec 12), (6 \prec 12)}](https://tex.z-dn.net/?f=A%3D%7B%282%20%5Cprec%204%29%2C%20%282%20%5Cprec%206%29%2C%20%282%20%5Cprec%208%29%2C%20%282%20%5Cprec%2010%29%2C%20%282%20%5Cprec%2012%29%2C%20%283%20%5Cprec%206%29%2C%20%283%20%5Cprec%209%29%2C%20%284%20%5Cprec%208%29%2C%20%284%20%5Cprec%2012%29%2C%20%286%20%5Cprec%2012%29%7D)
Hence the required Hasse diagram is shown in figure 1:
In the Hasse diagram 2 and 3 are on the same level as they are not related.
The next numbers are 4, 6, 9, and 10. 4, 6 and 10 are divisible by both 2. 6 and 9 are divisible by 3. Then 8 and 12 are divisible by 4 also 12 is divisible by 6.
Hence, the required diagram of the partial order of the set {2,3,4,6,8,9,10,12} is shown in figure 1.