The integers satisfy a property known as mathematical induction. This is a familiar topic in high...

The integers satisfy a property known as mathematical induction. This is a familiar topic in high school textbooks.

(a) The First Principle of Mathematical Induction is stated as follows. Suppose S is a subset of N with the following properties: (i) The number 1 is in S. (ii) If n is in S, then n + 1 is in S. Using well-ordering, prove S = N.

(b) The Second Principle of Mathematical Induction is stated as follows. Suppose S is a subset of N with the following properties: (i) The number 1 is in S. (ii) If n is in S and if every natural number k, where k ≤ n, is in S, then n + 1 is in S. Using well-ordering, prove S = N.

Show, using mathematical induction, that 2k−1 ≤ k! for all natural numbers k, by doing parts (a) and (b).

(a) Let S be the set of natural numbers for which the inequality is true. Show that 1 is in S.

(b) Now conclude, using the induction hypothesis and the theorems about inequalities, that 2 n = 2 · 2 n−1 ≤ 2 · n! ≤ (n + 1) · n! = (n + 1)! That is, if n is in S, then n + 1 is in S.

Expert Solution

Colby Messinger answered 2 years ago

Let d1, d2, ..., dn, with n at least 2, be positive integers. Use mathematical induction...

Let d1, d2, ..., dn, with n at least 2, be positive integers. Use mathematical induction to explain why, if d1+ d2+…+dn = 2n-2, then there must be a tree with n vertices whose degrees are exactly d1, d2, ..., dn. (Be careful with reading this statement. It is not the same as saying that any tree with vertex degrees d1, d2, ..., dn must satisfy d1+ d2+...+dn = 2n-2, although this is also true. Rather, it says that if...

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

What is similarity and difference between first principle induction mathematical and second principle induction mathematical ?...

What is similarity and difference between first principle induction mathematical and second principle induction mathematical ? When to use each principle? are there characteristics that distinguish the issue to be solved by the first or second principle?!

Prove that the proof by mathematical induction and the proof by strong 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.

full theory and mathematical derivatives behind induction type Instruments

1.create and solve Example of Mathematical induction uses in real life.

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

Do the following problems: a. Use mathematical induction to show that 1 + 2 + 22...

Do the following problems: a. Use mathematical induction to show that 1 + 2 + 22 +···+ 2n = 2n+1 − 1 for all non-negative integers n. b. A coin is weighted so that P(H) = 2/3 and P(T) = 1/3. The coin is tossed 4 times. Let the random variable X denote the number of heads that appear. (x) Find the distribution of X; (xx) Find the expectation E(X). c. Show a derivation of Bayes’ Theorem

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.

Question