COMP232 - Mathematics for Computer Science

Transcription

1 COMP232 - Mathematics for Computer Science Tutorial 9 Ali Moallemi moa ali@encs.concordia.ca Iraj Hedayati h iraj@encs.concordia.ca Concordia University, Winter 2017 Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 1 / 8

2 Table of Contents Strong Induction andwell-ordering Exercise 4 Exercise 6 Exercise 12 Exercise 26 Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 2 / 8

3 Exercise 4 Let P(n) be the statement that a postage of n cents can be formed using just 4-cent stamps and 7-cent stamps. The parts of this exercise outline a strong induction proof that P(n) is true for n 18. a) Show that the statements P(18), P(19), P(20), and P(21) are true, completing the basis step of the proof. answer: P(18) = , P(19) = , P(20) = , P(21) = , b) What is the inductive hypothesis of the proof? answer: The statement that using just 4-cent and 7-cent stamps we can form j cents postage for all j with 18 j k, where we assume that k 21 c) What do you need to prove in the inductive step? answer: Assuming the inductive hypothesis, we can form k + 1 cents postage using just 4-cent and 7-cent stamps. Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 3 / 8

4 Exercise 3 Cont... Let P(n) be the statement that a postage of n cents can be formed using just 4-cent stamps and 7-cent stamps. The parts of this exercise outline a strong induction proof that P(n) is true for n 18. d) Complete the inductive step for k 21. answer: Because k 21, we know that P(k 3) is true, that is, that we can form k 3 cents of postage. Put one more 4-cent stamp on the envelope, and we have formed k + 1 cents of postage. e) Explain why these steps show that this statement is true whenever n 18. answer: We have completed both the basis step and the inductive step, so by the principle of strong induction, the statement is true for every integer n greater than or equal to 18. Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 4 / 8

5 Exercise 6 a) Determine which amounts of postage can be formed using just 3-cent and 10-cent stamps. answer: {3, 6, 9, 10, 12, 13, 15, 16} {n : n 18} b) Prove your answer to (a) using the principle of mathematical induction. Be sure to state explicitly your inductive hypothesis in the inductive step. answer: P(18) = P(19) = P(20) = Now we show P(k + 1) can be constructed from P(k). -If it has two 10, then replace them with seven 3. -If it has One/ or no 10, it Should have at least three 3 (to be larger that 20), then replace them with a 10. Then P(k + 1) is built. Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 5 / 8

6 Exercise 6 cont... c) Prove your answer to (a) using strong induction. How does the inductive hypothesis in this proof differ from that in the inductive hypothesis for a proof using mathematical induction? answer: P(18) = P(19) = P(20) = Let P(j) be true for 18 j k where k 20, then we show P(k + 1) is also true. We know that P(k 2) is true since 18 k 2 k then P(k + 1) = P(k 2) + 3. Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 6 / 8

7 Exercise 12 Use strong induction to show that every positive integer n can be written as a sum of distinct powers of two, that is, as a sum of a subset of the integers 2 0 = 1, 2 1 = 2, 2 2 = 4 and so on. [Hint: For the inductive step, separately consider the case where k + 1 is even and where it is odd. When it is even, note that (k + 1)/2 is an integer.] answer: Clearly 2 0 = 1, 2 1 = 2. Let assume for any 1 j k where k 2 it is true. Then if k + 1 is even, we have (k + 1)/2 is integer and 1 (k + 1)/2 k, as a result P((k + 1)/2) is true. Then P(k + 1) = 2 P((k + 1)/2). Similarly if k + 1 is odd, we have k/2 is integer and 1 k/2 k, as a result P(k/2) is true. Then P(k + 1) = 2 P(k/2) Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 7 / 8

8 Exercise 26 Suppose that P(n) is a propositional function. Determine for which nonnegative integers n the statement P(n) must be true if a) P(0) is true; for all nonnegative integers n,if P(n) is true, then P(n + 2) is true. answer: p(2n) for all n 0. b) P(0) is true; for all nonnegative integers n,if P(n) is true, then P(n + 3) is true. answer: p(3n) for all n 0. c) P(0) and P(1) are true; for all nonnegative integers n,if P(n) and P(n + 1) are true, then P(n + 2) is true. answer: p(n) for all n 0. d) P(0) is true; for all nonnegative integers n,if P(n) is true, then P(n + 2) and P(n + 3) are true. answer: p(n) for all n 0 and n 1. Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 8 / 8