Prove that the proof by mathematical induction and the proof by strong induction are equivalent

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Colby Messinger answered 2 years ago

Course: Discrete Math Show/Prove that the principal of strong induction and regular induction are equivalent.

Provide an example of a proof by mathematical induction. Indicate whether the proof uses weak induction...

Provide an example of a proof by mathematical induction. Indicate whether the proof uses weak induction or strong induction. Clearly state the inductive hypothesis. Provide a justification at each step of the proof and highlight which step makes use of the inductive hypothesis.

Prove by strong mathematical induction that any integer greater than 1 is divisible by a prime...

Prove by strong mathematical induction that any integer greater than 1 is divisible by a prime number.

In this problem we prove that the Strong Induction Principle and Induction Principle are essentially equiv-...

In this problem we prove that the Strong Induction Principle and Induction Principle are essentially equiv- alent via Well-Ordering Principle. (a) Assume that (i) there is no positive integer less than 1, (ii) if n is a positive integer, there is no positive integer between n and n+1, and (iii) the Principle of Mathematical Induction is true. Prove the Well-Ordering Principle: If X is a nonempty set of positive integers, X contains a least element. (b) Assume the Well-Ordering Principle...

Create a mathematical proof to prove the following: Given an integer n, and a list of...

Create a mathematical proof to prove the following: Given an integer n, and a list of integers such that the numbers in the list sum up to n. Prove that the product of a list of numbers is maximized when all the numbers in that list are 3's, except for one of the numbers being either a 2 or 4, depending on the remainder of n when divided by 3.

Please be able to follow the COMMENT Use induction proof to prove that For all positive...

Please be able to follow the COMMENT Use induction proof to prove that For all positive integers n we have the inequality n<=2^n here is the step: base step: P(1)= 1<=2^1 inductive step: k+1<= 2^(k)+1 <= 2^(k)+k (since k>=1) <= 2^(k)+2^(k) = 2X2^(k) =2^(k+1) i don't understand why 1 can be replaced by k and i don't know why since k>=1

Prove using the principle of mathematical induction: (i) The number of diagonals of a convex polygon...

Prove using the principle of mathematical induction: (i) The number of diagonals of a convex polygon with n vertices is n(n − 3)/2, for n ≥ 4, (ii) 2n < n! for all n > k > 0, discover the value of k before doing induction

a. Use mathematical induction to prove that for any positive integer ?, 3 divide ?^3 +...

a. Use mathematical induction to prove that for any positive integer ?, 3 divide ?^3 + 2? (leaving no remainder). Hint: you may want to use the formula: (? + ?)^3= ?^3 + 3?^2 * b + 3??^2 + ?^3. b. Use strong induction to prove that any positive integer ? (? ≥ 2) can be written as a product of primes.

Prove these scenarios by mathematical induction: (1) Prove n2 < 2n for all integers n>4 (2)...

Prove these scenarios by mathematical induction: (1) Prove n2 < 2n for all integers n>4 (2) Prove that a finite set with n elements has 2n subsets (3) Prove that every amount of postage of 12 cents or more can be formed using just 4-cent and 5-cent stamps

If you prove by strong induction a statement of the form ∀ n ≥ 1P(n), the...

If you prove by strong induction a statement of the form ∀ n ≥ 1P(n), the inductive step proves the following implications (multiple correct answers are possible): a) (P(1) ∧ P(2)) => P(3) b) (P(1) ∧ P(2) ∧ P(3)) => P(4) c) P(1) => P(2)

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