1. Some proper partitions P1,…,Pn of a finite set S (that is, partitions containing at least two parts) are called independent if no matter how we choose one class from each partition, the intersection of the chosen classes is nonempty. Show that if the inequality\frac{\left | S \right | }{2} < \left |P_1 \right | \dots \left |P_n \right |\qquad (*)holds for some independent partitions, then P1,…,Pn is maximal in the sense that there is no partition P such that P,P1,…,Pn are independent. On the other hand, show that inequality (∗) is not necessary for this maximality. (C.20)
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