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4. Let Yn be the set of compositions of n all of whose parts are equal to 1 or 2. For example, Yi = {(1)} and Y2 = {(1, 1), (2)}. Let Y, be the subset of Y, consisting of all of the elements of Y, whose last part is 1. Let Y," be the subset of Y, consisting of all of the elements of Y, whose last part is 2. Show: (a) |Yal = 1Yl + 1Y,"'1. (b) [Yil = \\Yn-1/ and |Y,"'| = |Yn-21. (c) Recall the Fibonacci sequence (fn)n>1 is defined recursively by f1 = f2 = 1 and fn+1 = fn + fn-1. Use (a), (b) and Mathematical Induction to prove that |Ym| = fn+1 for n 2 1