Suppose we want to prove: “If \( n \) is an even natural number, then for all \( x \in \mathbb{R} \), \( x^n \geq 0 \).”Classify each of the following as either a mistake or the beginning of a direct proof, proof by contrapositive, or proof by contradiction.1. Let \( n \) be an even natural number. Suppose there exists an \( x \in \mathbb{R} \) such that \( x^n < 0 \).2. Let \( n \) be an odd natural number, and suppose that for all \( x \in \mathbb{R} \), \( x^n \geq 0 \).3. Let \( n \) be an even natural number. Suppose that for all \( x \in \mathbb{R} \), \( x^n < 0 \).4. Take an arbitrary \( n \in \mathbb{N} \). Suppose there exists an \( x \in \mathbb{R} \) such that \( x^n < 0 \).5. Let \( n \) be any odd natural number.

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Answered: Suppose we want to prove: "If n is an…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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"for all natural numbers m, n, p, we have that if p | (m * n) then either p | m or p | n" Try different concrete values for m, n, and p. Is this a true statement or a false statement? If false, give a counterexample. If true, find a proof.
a. #5 Prove that for all integers n, it is the case that n is even if an only if 3n is even. That is, prove both implications: if n is even, then 3n is even, and if 3n is even, then n is even. b. #7 Consider the statement: for all integers a and b, if a is even and b is a multiple of 3, then ab is a multiple of 6.Then state the converse, tell if it is true, and prove or disprove.
In chapter 1.1 “Elementary Number Theory & Its Applications” by Kenneth H. Rosen, I want to prove or disprove the following: Exercise 4a) The sum of a rational number and an irrational number is irrational. The book states about rational numbers: “The real number r is rational if there are integers p and q, with q not equaling zero such that r=p/q. If r is not rational, it is said to be irrational.”
Question 2 Write a formal proof for each of the following claims; some of them were used in class. In what follows, K, a, b are positive integers. In each part you may use the previous parts of the question, even if you did not solve it. a) If b+ 0 and a | b then a 1 such that a b then a K. Hint: Use proof by contradiction. That is, assume that both a b and a | K hold and then use Part a) to reach a contradiction.
Suppose A is a set of real numbers such that 7∈A and supA >7. Must A contain a number greater than 7? Either prove that it must or give a counter-example.
The Archimedean Property states that the set N of natural numbers is unbounded above in R.It is equivalent to the following. (a) For each z ∈ R, there exists an n ∈ N such that n > z.(b) For each x > 0 and for each y ∈ R, there exists an n ∈ N such that nx > y.(c) For each x > 0, there exists an n ∈ N such that 0 <1/n < x.
Prove each of the following statements using a direct proof, a proof by contrapositive, or a proof by contradiction (and cases when needed). For each statement, indicate which proof method you used, as well as the assumptions (what you suppose) and the conclusions (what you need to show) of the proof.A.) For any integers x and y, if x + y > 50, then x > 20 or y > 30.B.) For any integers x, y, and z with x ≠ y, if z is divisible by x − y, then z is divisible by y − x.C.) The difference of any rational number and any irrational number is irrational.

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