Define an integer k to be odd if k - 1 is even. Write up a formal proof of the following, usingan indirect proof:Claim: For any natural numbers m and n, if m is odd and n is odd, then mn is even.(Notice--this is not hard to prove, but make sure that you are using a proof by contradiction.And use the official definitions of even and odd.)

Claim: For any natural numbers m and n, if m is odd and n is odd, then m+n is even.To prove the…

Answered: Define an integer k to be odd if k - 1…

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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Prove the following using direct proof.a. If 5? − 3 = ? + 5, then ? = 2.b. If ? and ? are both odd, then ? + ? is even.c. The sum of two odd integers is even.2. Prove the following using proof by contrapositive.a. If ? and ? are even integers, then ? + ? is an even integer.b. If ? and ? are odd integers, then ?? is an odd integer.c. If ? is an odd integer, then ?3 + ? is even.3. Prove the following using proof by contradiction.a. If ?2is an even integer, then ? is an even integer.b. If ? and ? are odd integers, then ? + ? is an even integer.c. Prove that √2 is irrational.
n > 1 is an integer. Which of the following statements is false? Select one: a. If 10" – 1 is prime then n is prime. b. If n is even then 2" + 1 is prime. c. If 2" – 1 is prime then n is prime. d. If n is composite then 10" – 1 is composite. e. If n is composite then 2" – 1 is composite.
"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.
Prove or disprove the following statement, (c):
- A Pythagorean prime number is a prime number of the form 4n + 1, where n ≥ 1. A Mersenne prime number is a prime number of the form 2" – 1. A perfect number is a number that is equal to the sum of all of its divisors, including 1 but excluding the number itself. Consider the following statements: (1) Let m is the smallest Pythagorean prime number. Then, 2¹ – 1 is a Mersenne prime number. (2) If m is a perfect number, then 2 – 1 is a Mersenne prime number. Select one of the following choices: (a) (1) is True and (2) is False. (c) (1) and (2) are True. (b) (1) is False and (2) is True. (d) (1) and (2) are False.
Give a carefully worded proof for the following statements using direct proof, contrapositive proof or proof by contradiction. No matter which proof method you choose to use, make sure you follow the outline properly. (If-and-Only-If Proof) Given an integer a and b, then b2(a + 3) is even if and only if a is odd or b is even. (Use the basic definitions of even and odd in your proof.)
In each of the following statements, n represents a positive integer. One of the statements is true and the other two are false. If n is even, then n is not prime if n is a square number, then n is not prime if n is a prime number and n ≥ 10, then n+2 is prime or n+4 is prime. A. What two statements are false and in each case give a counter example to show that it is false B. Prove the true statement
Use the PMI to prove that 3 - 1 is even for all n E N. Proof. Base case: Since 3¹ - 1 = 2, which is even. Thus the statement is true for n = [Select] Inductive step: Assume that there is a natural number n such that 3 - 1 is even. Then 3-1 = [Select] for some integer k. Then 3" [Select] ✓. On both sides, first multiply by 3, then subtract 1, and simplify to get 3+1 -1 = 2( [Select] which is an [Select] integer. Hence, by the PMI, 3 - 1 is even for all n E N. )
Give a carefully worded proof for the following statements using direct proof, contrapositive proof or proof by contradiction. No matter which proof method you choose to use, make sure you follow the outline properly. For every integer n, 4 ∤ (n2 - 3). (reads “4 does not divide (n2 - 3)).
Give a direct proof: Let a, b, and c be integers. If a | b and a c, then a | (b. c). Remember that you must use the definition of | in your proof. Suppose a, b, and care integers and a | b and a | c. By the definition of 1, b = ak₁ and c = for some integers k₁ and k₂. Therefore, b . c = a, and is an integer, so a | b. c.
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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