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\$$?