1) Write down all the partitions of a two element set {\\(\bullet,\ast\\)}, order them as above, and draw the Hasse diagram.

2) Now do the same thing for a four element-set, say {1,2,3,4}. There should be 15 partitions.

Choose any two systems in your 15-element Hasse diagram, call them \\(A\\) and \\(B\\).

3) What is \\(A\vee B\\), using the definition given in the paragraph above

4) Is it true that \\(A\leq (A\vee B)\\) and \\(B\leq (A\vee B)\\)?

5) What are all the partitions \\(C\\) for which both \\(A\leq C\\) and \\(B\leq C\\).

6) Is it true that in each case, \\((A\vee B)\leq C\\)?

2) Now do the same thing for a four element-set, say {1,2,3,4}. There should be 15 partitions.

Choose any two systems in your 15-element Hasse diagram, call them \\(A\\) and \\(B\\).

3) What is \\(A\vee B\\), using the definition given in the paragraph above

4) Is it true that \\(A\leq (A\vee B)\\) and \\(B\leq (A\vee B)\\)?

5) What are all the partitions \\(C\\) for which both \\(A\leq C\\) and \\(B\leq C\\).

6) Is it true that in each case, \\((A\vee B)\leq C\\)?