PMI Principle of Mathematical Induction: If SCN and 0 € S and (k+1) € S wheneverkES, then S=N.WOP Well-Orderering Property: N is well-ordered by the relation <.Show that if WOP holds for N then PMI also holds for N. Hint: Follow the idea given in theproof from the lecture showing that if PMI is true then WOP holds, for all subsets of N in theform S = {n EN:P (n)}, where P (n) is a first order formula.Start arguing by contradiction: Suppose that SCN, 0 ES and (k+1) ES whenever k E Sand S‡N.• Define T=M\S and explain why T # 0.• Explain why mo = min (T) exists and argue that mō > 0.. Consider k= (mo-1) and argue that k E T. Then explain why k E S.• What do you know about S which you can use to conclude that (k+1) € S. Then show thatk+1=mo.. Conclude that mo E S. Explain why does this yield a contradiction.• Finish your argument.

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Answered: Show that if WOP holds for N then PMI…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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4. Please
5.3.6 Here is a 'proof' that all human beings have the same age. Where is the flaw in the argu- ment? Proof. (Base case n = 1) In a set with only 1 person, all the people in the set have the same age. (Inductive hypothesis) Suppose that for some integer n > 1 and for all sets with n people, it is true that all of the people in the set have the same age. (Inductive step) Let A be a set with n +1 people, say A = {a1,...,an, an+1}, and let A' = {a1,... , an} and A" = {a2,. , an+1}. The inductive hypothesis tells us that all the people in A' have the same age and all the people in A" have the same age. Since a2 belongs to both sets, then all the people in A have the same age as a2. We conclude that all the people in A have the same age. (Conclusion) By induction, the claim holds for all n > 1.
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Resource Allocation You manage an ice cream factory that makes two flavors: Creamy Vanilla and Continental Mocha. Into each quart of Creamy Vanilla go 1 egg and 2 cups of cream. Into each quart of Continental Mocha go 2 eggs and 1 cup of cream. You have in stock 700 eggs and 800 cups of cream. Draw the feasible region showing the number of quarts of vanilla and number of quarts of mocha that can be produced. (Place Creamy Vanilla on the x-axis and Continental Mocha on the y-axis.) The x y-coordinate plane is given. There are 2 lines and a region labeled "solution set" on the graph. The first line enters the window in the second quadrant, goes down and right, passes through the approximate point (0, 350), passes through the point (300, 200) crossing the second line, crosses the x-axisat x = 700, and exits the window in the fourth quadrant. The second line enters the window in the second quadrant, goes down and right, passes through the approximate point (0, 800), passes through…
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Exercise 1. Prove that for all n = Z, if nmod4 = 1 or nmod4 = 3, then 8|n² — 1. File Preview

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