A3.1 Mathematical Induction direct proof indirect proof (proof by - PowerPoint PPT Presentation

Discrete Mathematics in Computer Science September 23, 2020 — A3. Proofs II Discrete Mathematics in Computer Science A3. Proofs II A3.1 Mathematical Induction Malte Helmert, Gabriele R¨ oger A3.2 Structural Induction University of Basel September 23, 2020 Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science September 23, 2020 1 / 21 Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science September 23, 2020 2 / 21 A3. Proofs II Mathematical Induction A3. Proofs II Mathematical Induction Proof Techniques most common proof techniques: A3.1 Mathematical Induction ◮ direct proof ◮ indirect proof (proof by contradiction) ◮ contrapositive ◮ mathematical induction ◮ structural induction Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science September 23, 2020 3 / 21 Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science September 23, 2020 4 / 21

A3. Proofs II Mathematical Induction A3. Proofs II Mathematical Induction Mathematical Induction Propositions Consider a statement on all natural numbers n with n ≥ m . ◮ E.g. “Every natural number n ≥ 2 can be written as a product Concrete Mathematics by Graham, Knuth and Patashnik (p. 3) of prime numbers.” Mathematical induction proves that ◮ P (2): “2 can be written as a product of prime numbers.” ◮ P (3): “3 can be written as a product of prime numbers.” we can climb as high as we like on a ladder, ◮ P (4): “4 can be written as a product of prime numbers.” by proving that we can climb onto the bottom rung (the basis) ◮ . . . ◮ P ( n ): “ n can be written as a product of prime numbers.” and that ◮ For every natural number n ≥ 2 proposition P ( n ) is true. from each rung we can climb up to the next one (the step). A proposition P ( n ) is a mathematical statement that is defined in terms of natural number n . Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science September 23, 2020 5 / 21 Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science September 23, 2020 6 / 21 A3. Proofs II Mathematical Induction A3. Proofs II Mathematical Induction Mathematical Induction Mathematical Induction: Example I Theorem For all n ∈ N 0 with n ≥ 1 : � n k =1 (2 k − 1) = n 2 Mathematical Induction Proof. Proof (of the truth) of proposition P ( n ) Mathematical induction over n : for all natural numbers n with n ≥ m : basis n = 1: � 1 k =1 (2 k − 1) = 2 − 1 = 1 = 1 2 ◮ basis: proof of P ( m ) k =1 (2 k − 1) = m 2 for all 1 ≤ m ≤ n IH: � m ◮ induction hypothesis (IH): inductive step n → n + 1: suppose that P ( k ) is true for all k with m ≤ k ≤ n ◮ inductive step: proof of P ( n + 1) � n +1 � � n � k =1 (2 k − 1) = k =1 (2 k − 1) + 2( n + 1) − 1 using the induction hypothesis = n 2 + 2( n + 1) − 1 IH = n 2 + 2 n + 1 = ( n + 1) 2 Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science September 23, 2020 7 / 21 Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science September 23, 2020 8 / 21

A3. Proofs II Mathematical Induction A3. Proofs II Mathematical Induction Mathematical Induction: Example II Mathematical Induction: Example II Theorem Theorem Every natural number n ≥ 2 can be written as a product of prime Every natural number n ≥ 2 can be written as a product of prime numbers, i. e. n = p 1 · p 2 · . . . · p m with prime numbers p 1 , . . . , p m . numbers, i. e. n = p 1 · p 2 · . . . · p m with prime numbers p 1 , . . . , p m . Proof. Proof (continued). Mathematical Induction over n : inductive step n → n + 1: ◮ Case 1: n + 1 is a prime number � trivial basis n = 2: trivially satisfied, since 2 is prime ◮ Case 2: n + 1 is not a prime number. IH: Every natural number k with 2 ≤ k ≤ n There are natural numbers 2 ≤ q , r ≤ n with n + 1 = q · r . IH: can be written as a product of prime numbers. . . . Using IH shows that there are prime numbers q 1 , . . . , q s with q = q 1 · . . . · q s and r 1 , . . . , r t with r = r 1 · . . . · r t . Together this means n + 1 = q 1 · . . . · q s · r 1 · . . . · r t . Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science September 23, 2020 9 / 21 Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science September 23, 2020 10 / 21 A3. Proofs II Mathematical Induction A3. Proofs II Mathematical Induction Weak vs. Strong Induction Is Strong Induction More Powerful than Weak Induction? Are there statements that we can prove with strong induction ◮ Weak induction: Induction hypothesis only supposes but not with weak induction? that P ( k ) is true for k = n ◮ Strong induction: Induction hypothesis supposes We can always use a stronger proposition: that P ( k ) is true for all k ∈ N 0 with m ≤ k ≤ n ◮ also: complete induction ◮ “Every n ∈ N 0 with n ≥ 2 can be written as a product of prime numbers.” Our previous definition corresponds to strong induction. ◮ P ( n ): “ n can be written as a product of prime numbers.” ◮ P ′ ( n ): “all k ∈ N 0 with 2 ≤ k ≤ n can be written Which of the examples had also worked with weak induction? P ′ ( n ) “ as a product of prime numbers.” Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science September 23, 2020 11 / 21 Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science September 23, 2020 12 / 21

A3. Proofs II Mathematical Induction A3. Proofs II Structural Induction Reformulating Statements It is sometimes convenient to rephrase a statement. For example: ◮ “7 n + 3 n is divisible by 10 for all odd n ∈ N 0 .” A3.2 Structural Induction ◮ “For all n ∈ N 0 : if n is odd then 7 n + 3 n is divisible by 10.” ◮ P ( n ) = “if n is odd then 7 n + 3 n is divisible by 10.” ◮ Need two base cases. ◮ Case distinction ( n even or odd) in inductive step ◮ “For all n ∈ N 0 : 7 (2 n +1) + 3 (2 n +1) is divisible by 10.” ◮ P ′ ( n ) = “7 (2 n +1) + 3 (2 n +1) is divisible by 10.” Be careful about how to reformulate a statement! Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science September 23, 2020 13 / 21 Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science September 23, 2020 14 / 21 A3. Proofs II Structural Induction A3. Proofs II Structural Induction Inductively Defined Sets: Examples Inductive Definition of a Set Example (Natural Numbers) The set N 0 of natural numbers is inductively defined as follows: ◮ 0 is a natural number. Inductive Definition ◮ If n is a natural number, then n + 1 is a natural number. A set M can be defined inductively by specifying ◮ basic elements that are contained in M Example (Binary Tree) ◮ construction rules of the form The set B of binary trees is inductively defined as follows: “Given some elements of M , another element of M ◮ � is a binary tree (a leaf) can be constructed like this.” ◮ If L and R are binary trees, then � L , � , R � is a binary tree (with inner node � ). Implicit statement: all elements of the set can be constructed Implicit statement: by finite application of these rules Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science September 23, 2020 15 / 21 Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science September 23, 2020 16 / 21