How to construct a class diagram?

Construct the Hasse diagram for the divides relation on the set {2, ... , 10}. Identify the minimal elements?

• Construct the Hasse diagram for the divides relation on the set {2, ... , 10}. Identify the minimal elements and the maximal elements.

• Answer:

  Write the number 8 on one line of a piece of paper. Write 9, 6, 4, and 10 on the next line of the same piece of paper. Write 3, 2, 5, and 7 on the line under that one. Draw line segments from 4 to 8, from 3 to 9, from 3 to 6, from 2 to 6, from 2 to 4, from 2 to 10, and from 5 to 10. 3, 2, 5, and 7 are the minimal elements (since there is nothing "below" them in the diagram). 9, 6, 8, 10, and 7 are the maximal elements (since there is nothing "above" them in the diagram). The explicit ordering of these numbers in each line of the paper is immaterial. I chose that ordering just so you can draw the picture cleanly. The essential features of the diagram: (1) each number is "below" the things it divides and "above" the things it is a multiple of, (2) there is a line from x to y whenever it is the case that (a) x divides y, and (b) there is no number z unequal to either x or y, with the property that both x divides z, and z divides y. The property (2) explains why there is no line from 2 to 8, for example. It is true that 2 divides 8, but there is something "between" 2 and 8 in this ordering, namely 4. We have that 2 divides 4, and 4 divides 8. So we do not put a line between 2 and 8. The fact that 2 divides 8 is then seen in the diagram from the fact that there is a two-line chain from 2 up to 8.