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Suppose that the set S has n elements and discuss the number of subsets of various...

  1. Suppose that the set S has n elements and discuss the number of subsets of various sizes.
    (a) How many subsets of size 0 does S have?
    (b) How many subsets of size 1 does S have?
    (c) How many subsets of size 2 does S have?
    (d) How many subsets of size n does S have?
    (e) Clearly the total number of subsets of S must equal the sum of the number of subsets of size 0, of size 1, of size 2, . . . , and of size n. Write an equation that equates the sum of the number of subsets of these various sizes with the total number of subsets of a set of n elements.
    (f) Apply your equation from (e) to the case n = 6. In other words, write an equation that has the sum of the number of subsets of sizes 0, 1, 2, 3, 4, 5, and 6 on one side and the total number of subsets on the other side.
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