Question

(a) Use induction to prove that, for any integer (n geq 0), the set ({x_1, x_2, ..., x_n}) has exactly (2^n) different subsets. Notice that when (n=0), the set ({x_1, x_2, ..., x_n} = emptyset). Hint: in the induction step, consider the mutually exclusive cases of subsets that contain (x_n), and subsets that do not contain (x_n). (b) Explain how to modify the argument in (a) to obtain a proof by induction that, for any integer (n geq 0), the number of sequences of 0s and 1s with length (n) is exactly (2^n). (c) Explain how to modify the argument above to prove that, for any integer (n geq 0), the number of sequences of 0s, 1s, and 2s with length (n) is exactly (3^n). (d) State a general result based on parts (b) and (c). No proof is needed.

          (a) Use induction to prove that, for any integer (n geq 0), the set ({x_1, x_2, ..., x_n}) has exactly (2^n) different subsets.
Notice that when (n=0), the set ({x_1, x_2, ..., x_n} = emptyset). Hint: in the induction step, consider the mutually exclusive cases of subsets that contain (x_n), and subsets that do not contain (x_n).

(b) Explain how to modify the argument in (a) to obtain a proof by induction that, for any integer (n geq 0), the number of sequences of 0s and 1s with length (n) is exactly (2^n).

(c) Explain how to modify the argument above to prove that, for any integer (n geq 0), the number of sequences of 0s, 1s, and 2s with length (n) is exactly (3^n).

(d) State a general result based on parts (b) and (c). No proof is needed.

Added by Encarnacion P.

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