1. Prove the following propositions:(a) For all real numbers x and y, if x is rational and y is irrational,then x+y is irrational.(b) For all nonzero real numbers x and y, if x is rational and y isirrational, then x/y is irrational.2. If n is a positive integer, then n4 - n is divisible by 4. [Proof ofExhaustion]3. There are no positive integers a and b such that 18a + 6b = 1.(a) Write this statement as: if P then 18a + 6b 1.(b) What are the first two sentences of your contradiction proofof this claim?(c) Find a contradiction.(d) Conclude the proof.

QUESTION 1 a. The sum of a rational number plus an irrational number is always irrational, therefore…

1. Prove the following propositions: (a) For all real numbers x and y, if x is rational and y is irrational, then x+y is irrational. (b) For all nonzero real numbers x and y, if x is rational and y is irrational, then x/y is irrational. 2. If n is a positive integer, then nª-n is divisible by 4. [Proof of Exhaustion] 3. There are no positive integers a and b such that 18a + 6b = 1. (a) Write this statement as: if P then 18a + 6b/ 1. (b) What are the first two sentences of your contradiction proof of this claim? (c) Find a contradiction. (d) Conclude the proof.

1. Prove the following propositions: (a) For all real numbers x and y, if x is rational and y is irrational, then x+y is irrational. (b) For all nonzero real numbers x and y, if x is rational and y is irrational, then x/y is irrational. 2. If n is a positive integer, then nª-n is divisible by 4. [Proof of Exhaustion] 3. There are no positive integers a and b such that 18a + 6b = 1. (a) Write this statement as: if P then 18a + 6b/ 1. (b) What are the first two sentences of your contradiction proof of this claim? (c) Find a contradiction. (d) Conclude the proof.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

See similar textbooks

Similar questions

1) Direct proofs: A. Prove that for any integer x, the integer x(x + 1) is even. B. Prove that n is even if and only if n^2 is even.
Problem 2: Consider the set of integers. 1. Use a direct proof to show that the sum of two odd integers is even. 2. Use a direct proof to show that the product of two odd integers is odd. Prove that if n is an integer and n³ + 5 is odd, then n is even using a) A proof by contraposition. 3. b) A proof by contradiction. 4. Prove or disprove that if m and n are integers such that mn = 1, then either m = 1 and n = 1, or else m = -1 and n = -1. 5. Prove that these four statements about the integer n are equivalent: a) n² is odd. b) 1 - n is even. c) n³ is odd. d) n² + 1 is even. 6. Prove that if m and n are integers and mn is even, then m is even or n is even.
Prove or disprove . show your work . (a) for any integers n a and m : if both n and m are odd, then n – m? is even (b) Vp € Z+: if p is prime, then p- 2 is not prime. (c) Vs ER : s is irrational s2 is irrational. (d) There is two odd integers n and m such that n2 – m2 – 1 is odd.
A prime number is a natural number greater than 1 which is not a product of two smaller natural numbers. Prove or disprove: For every integer q, if q > 7, then q can be written as q = (a + b * c) such that the following properties hold a and b are prime numbers, c is an integer greater than 1.
METHODS OF PROOF PROVE THE FOLLOWING: 1. By contradiction, prove that "for all real numbers x, if x is irrational, then -x is irrational. *Please provide a clear proof. THANK YOU.
5. Prove algebraically, the following: a) (A + B')(A' + B) = AB + A'B' b) AB' + AB + A' = 1 Use only basic theorems to prove them.
Fourth Question: Use rules of inference to show that the hypotheses "If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on," “If the sailing race is held, then the trophy will be awarded," and "The trophy was not awarded" imply the conclusion "It rained.". Fifth Question: Prove that if n is an integer and 3n + 2 is even, then n is even using (a) a proof by contraposition. (b) a proof by contradiction.
Determine which of the following arguments is valid and which is invalid. Justify your answer using universal modus ponens or universal modus tollens (for the valid argument), and the converse or the inverse error (for the invalid argument). (a) If an infinite series converges, then the terms go to 0. The terms of the infinite series 1/ go to 0. :. The infinite series converges. n=1 (b) If an infinite series converges, then its terms go to 0. The terms of the infinite series .. The infinite series 11 do not go to 0. 11 does not converge.
Note: For all integers k,n it is true that kn, k+n, and k-n are integers. An integer k is even if and only if there exists an integer r such that k=2r. An integer k is odd if and only if there exists an integer r such that k=2r+1. For every integer k it is true that if k is even then k is not odd. For every integer k it is true that if k is odd then k is not even. For every integer k it is true that if k is not even then k is odd. For every integer k it is true that if k is not odd then k is even. If P then Q means the same thing as P → Q (P implies Q). 3. Consider the argument form pvr .. p→r Is this argument form valid? Prove that your answer is correct. 4. Prove that for every integer d, if d³ is odd then d is odd.
Determine whether the following statements are true or false. Justify your answer with a proof or a counterexample. a. For all integers , a,b,c if a I bc then a I b or a I c b. Determine the prime factorization of 44,574.
Prove that if n is a prime number, then √n is an irrational number. (The following theorem would be useful in your proof: Theorem: Let q, r be integers and p a prime number. If p | qr, then p | qor pr.)
I. Fermat's test consists of using Fermat's Theorem (an¹=1 (mod n)) with prime n. Just pick any number a E {1, ..., n-1} and check if it satisfies the expression. If not, then it must be a composite number, that is, it is not prime. II. Although Fermat's Theorem is used to check whether a number is prime or not, we can still obtain equality (an¹-1 (mod n)) even when n is not prime. E The numbers for which this situation occurs are called Carmichael numbers or pseudoprimes. III. Both the Miller-Rabin and AKS algorithms (checking a prime number) work for any type of input and are polynomial time. However, only AKS is deterministic. IV. The Miller-Rabin algorithm guarantees when a number is prime, but when it is composite (non-prime) it can only guarantee this characteristic in probabilistic terms. Only the following items are correct: A I, II, III B I, II, IV C I, III, IV D II, III, IV E I, II, III, IV