Strong Induction - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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CSE115/ENGR160 Discrete Mathematics

5.2 Strong induction and well-

ordering

• Use strong induction to show that if n is an

integer greater than 1, then n can be written

as the product of primes

• Let p(n) be the proposition that n can be

written as the product of primes

• Basis step: p(2) is true as 2 can be written as

the product of one prime, itself

• Inductive step: Assume p(k) is true with the

assumption that p(j) is true for j≤k

Proof with strong induction

• If k+1 is composite and can be written as a

product of two positive integers a and b, with

2≤a≤b<k+

• By inductive hypothesis, both a and b can be

written as product of primes

• Thus, if k+1 is composite, it can be written as

the product of primes, namely, the primes in

the factorization of a and those in the

factorization of b

Proof with induction

• Prove that every amount of postage of 12 cents or

more can be formed using just 4-cent and 5-cent

stamps

• First use mathematical induction for proof
• Basis step: Postage of 12 cents can be formed using 3

4-cent stamps

• Inductive step: The inductive hypothesis assumes

p(k) is true

• That is, we need to sure p(k+1) is true when k≥

Proof with strong induction

• Use strong induction for proof
• In the basis step, we show that p(12), p(13), p(14)

and p(15) are true

• In the inductive step, we show that how to get

postage of k+1 cents for k≥15 from postage of k-

cents

• Basis step: we can form postage of 12, 13, 14, 15

cents using 3 4-cent stamps, 2 4-cent/1 5-cent

stamps, 2 5-cent/1 4-cent stamps, and 3 5-cent

stamps. So p(12), p(13), p(14), p(15) are true

Proof with strong induction

• Inductive step: The inductive hypothesis is the statement p(j) is true for 12≤j ≤k, where k is an integer with k≥15. We need to show p(k+1) is true
• We can assume p(k-3) is true because k-3 ≥12, that is, we can form postage of k-3 cents using just 4-cent and 5-cent stamps
• To form postage of k+1 cents, we need only add another 4- cent stamp to the stamps we used to form postage of k- cents. That is, we show p(k+1) is true
• As we have completed basis and inductive steps of a strong induction, we show that p(n) is true for n≥
• There are other ways to prove this