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# Exercise 28 - Chapter 1

Let $$X$$ be the set of partitions of {•,◦,∗}; it has five elements and an order by coarseness, as shown in the Hasse diagram Eq. (1.2). Write down every pair $$(x,y)$$ of elements in $$X$$ such that $$x≤y$$. There should be 12.

$$(x2, x1) , (x3, x1), (x4, x1), (x5, x1), (x5, x2), (x5, x3), (x5, x4), (x1, x1), (x2, x2), (x3, x3), (x4, x4), (x5, x5)$$
Are the arrows backwards? Is $$\text{ more coarse } \le \text{ more fine }$$ ?
Comment Source:![Figure](https://docs.google.com/drawings/d/e/2PACX-1vSHvXcWpaxv-VMyJY037GdPlW4NN85kZ6WWgzLbZ1VWlQJ0jZ0BjwZNmiCJ3Vv3J5acg31H1BpGYICR/pub?w=502&h=383) \$$(x2, x1) , (x3, x1), (x4, x1), (x5, x1), (x5, x2), (x5, x3), (x5, x4), (x1, x1), (x2, x2), (x3, x3), (x4, x4), (x5, x5) \$$ Are the arrows backwards? Is \$$\text{ more coarse } \le \text{ more fine } \$$ ?