Exercise 8.6.1: Proofs by strong induction

Exercise 8.6.1: Proofs by strong induction - combining stamps. Prove each of the following statements using strong induction. (a) Prove that any amount of postage worth 8 cents or more can be made from 3-cent or 5-cent stamps. (b) Prove that any amount of postage worth 24 cents or more can be made from 7-cent or 5-cent stamps. (c) Prove that any amount of postage worth 12 cents or more can be made from 3-cent or 7-cent stamps.

Transcript

00:01 Hi, in this question we need to prove the result by mathematical induction.

00:05 Here, the amount of postage was 8 cents or more.

00:09 So let n greater than are equal to 8.

00:11 It made from 3 cents or 5 cents stamp.

00:15 So n equals 3a plus 5b, where a comma b belongs to natural number union 0.

00:21 Here, first step 1, consider n equals 8.

00:28 So that choose a equals 1.

00:32 And b equals 1 the 4 result whole for n equals 8 next assume n equals 9 here we need to take a equals 3 b equals 0 therefore n equals 9 that is 9 cent stamp can be made with 3 stamps made from 3 cents next n equals 10 so that choose a equals 0 b equals 2.

01:05 Therefore n equals 5 into 2 which is 10.

01:09 Therefore result hold for n equals 10.

01:14 Next move on to 2.

01:19 Here assume that the result hold for n equals j and j must be greater than or equal to 11.

01:31 So here j equals 3a1 plus 5 b 1 for some a a comma b, that is a1 comma b1 belongs to union of 0, that is natural number union of 0.

01:49 Next, move on to step 3.

01:50 Here we have to prove that n equals j plus 1.

01:54 So j plus 1 can be written as 3a1 plus 5 b1 plus b1 which is equal to 3a1 plus 5 b1 which is equal to 3a1 plus 5 b1 which is equal to 3 into a dash plus 5 b1 -b -b -dash.

02:10 A -dash and b -dash belongs to natural number union of 0.

02:13 Result hold for j plus 1.

02:15 Therefore, by induction hypothesis, hold for j plus 1 for all n greater than are equal to 8, n belongs to n union 0.

02:24 Next, in part 2, let n greater than are equal to 24.

02:28 We need to show n equals 5a plus 7b, a comma b belongs to natural number union of 0.

02:34 Here step 1, n equal 24, then a equals to b equals to result whole...

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