Let S be a given set. If, for some k > 0, S1, S2,......Sk are mutually exclusive nonemp...

Let S be a given set. If, for some k > 0, S1, S2,......Sk are mutually exclusive nonempty subsets of S such that

, then we call the set {S1,S2,... ,Sk} of a partition of S. Let Tn denote the number of different partitions of {1,2,.. n} Thus, T1 = 1 (the only partition being S1 = {1}) and T2= 2 (the two partitions being {{1,2,}}, {{1}, {2}}).

(a) Show, by computing all partitions, that T3 = 5, T4= 15.

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(b) Show that

and use this equation to compute T10