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Mastering Mathematical Induction in Discrete Structures

Learn the principle of mathematical induction through examples and proofs for all natural numbers, ideal for CMSC 203 course. Understand basis step, inductive step, and conclusion with practical applications in number theory. Explore Gauss' sum and prime factorization proofs. Discover the second principle of mathematical induction for advanced problem-solving. Enhance your understanding of induction techniques for tackling discrete structures effectively.

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Mastering Mathematical Induction in Discrete Structures

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  1. Follow me for a walk through... • Mathematical • Induction CMSC 203 - Discrete Structures

  2. Induction • The principle of mathematical induction is a useful tool for proving that a certain predicate is true for all natural numbers. • It cannot be used to discover theorems, but only to prove them. CMSC 203 - Discrete Structures

  3. Induction • If we have a propositional function P(n), and we want to prove that P(n) is true for any natural number n, we do the following: • Show that P(0) is true.(basis step) • Show that if P(n) then P(n + 1) for any nN.(inductive step) • Then P(n) must be true for any nN. (conclusion) CMSC 203 - Discrete Structures

  4. Induction • Example: • Show that n < 2n for all positive integers n. • Let P(n) be the proposition “n < 2n.” • 1. Show that P(1) is true.(basis step) • P(1) is true, because 1 < 21 = 2. CMSC 203 - Discrete Structures

  5. Induction • 2. Show that if P(n) is true, then P(n + 1) is true.(inductive step) • Assume that n < 2n is true. • We need to show that P(n + 1) is true, i.e. • n + 1 < 2n+1 • We start from n < 2n: • n + 1 < 2n + 1  2n + 2n = 2n+1 • Therefore, if n < 2n then n + 1 < 2n+1 CMSC 203 - Discrete Structures

  6. Induction • Then P(n) must be true for any positive integer.(conclusion) • n < 2n is true for any positive integer. • End of proof. CMSC 203 - Discrete Structures

  7. Induction • Another Example (“Gauss”): • 1 + 2 + … + n = n (n + 1)/2 • Show that P(0) is true.(basis step) • For n = 0 we get 0 = 0. True. CMSC 203 - Discrete Structures

  8. Induction • Show that if P(n) then P(n + 1) for any nN. (inductive step) • 1 + 2 + … + n = n (n + 1)/2 • 1 + 2 + … + n + (n + 1) = n (n + 1)/2 + (n + 1) • = (2n + 2 + n (n + 1))/2 • = (2n + 2 + n2 + n)/2 • = (2 + 3n + n2 )/2 • = (n + 1) (n + 2)/2 • = (n + 1) ((n + 1) + 1)/2 CMSC 203 - Discrete Structures

  9. Induction • Then P(n) must be true for any nN. (conclusion) • 1 + 2 + … + n = n (n + 1)/2 is true for all nN. • End of proof. CMSC 203 - Discrete Structures

  10. Induction • There is another proof technique that is very similar to the principle of mathematical induction. • It is called the second principle of mathematical induction. • It can be used to prove that a propositional function P(n) is true for any natural number n. CMSC 203 - Discrete Structures

  11. Induction • The second principle of mathematical induction: • Show that P(0) is true.(basis step) • Show that if P(0) and P(1) and … and P(n),then P(n + 1) for any nN.(inductive step) • Then P(n) must be true for any nN. (conclusion) CMSC 203 - Discrete Structures

  12. Induction • Example: Show that every integer greater than 1 can be written as the product of primes. • Show that P(2) is true.(basis step) • 2 is the product of one prime: itself. CMSC 203 - Discrete Structures

  13. Induction • Show that if P(2) and P(3) and … and P(n),then P(n + 1) for any nN. (inductive step) • Two possible cases: • If (n + 1) is prime, then obviously P(n + 1) is true. • If (n + 1) is composite, it can be written as the product of two integers a and b such that2  a  b < n + 1. • By the induction hypothesis, both a and b can be written as the product of primes. • Therefore, n + 1 = ab can be written as the product of primes. CMSC 203 - Discrete Structures

  14. Induction • Then P(n) must be true for any nN. (conclusion) • End of proof. • We have shown that every integer greater than 1 can be written as the product of primes. CMSC 203 - Discrete Structures

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