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When writing proofs by contradiction we begin by assuming the opposite of a statement, and then show that this leads to (i.e. entails) a contradiction. Which of the following statements constitutes a contradiction (that is, which of the following evaluates to false)? Suppose A and B to be sets of natural numbers, and x to be a natural numbers. Select all that apply. A. O (* E A) ^ (x ¢ B) ^ (æ € (AN B)) В. О (æ € B) → ((x € A) → (x € B)) ^ (x € A) C. O (x E A) ^ (x € B) ^ (AN B = 0) D. O (a E A) ^ (x e B) ^ (x ¢ (AU B)) E. O None of the above

When writing proofs by contradiction we begin by assuming the opposite of a statement, and then show that this leads to (i.e. entails) a contradiction. Which of the following statements constitutes a contradiction (that is, which of the following evaluates to false)? Suppose A and B to be sets of natural numbers, and x to be a natural numbers. Select all that apply. A. O (* E A) ^ (x ¢ B) ^ (æ € (AN B)) В. О (æ € B) → ((x € A) → (x € B)) ^ (x € A) C. O (x E A) ^ (x € B) ^ (AN B = 0) D. O (a E A) ^ (x e B) ^ (x ¢ (AU B)) E. O None of the above

Algebra: Structure And Method, Book 1
Algebra: Structure And Method, Book 1
(REV)00th Edition
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Chapter10: Inequalities
Section10.2: Solving Inequalities
Problem 65WE
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When writing proofs by contradiction we begin by assuming the opposite of a statement, and then show that this leads to (i.e. entails) a contradiction. Which of the following statements constitutes a contradiction (that is, which of the following evaluates to false)? Suppose A and B to be sets of natural numbers, and to be a natural numbers. Select all that apply. A. O (x E A) ^ (x ¢ B) ^ (x € (AN B)) В. О (хЕ В) — ((х E A) + (х € В))^ (х€ A) C. O (x E A) A (x € B) ^ (AN B = 0) D. O (x E A) ^ (x E B) ^ (x ¢ (AU B)) E. O None of the above
Transcribed Image Text:When writing proofs by contradiction we begin by assuming the opposite of a statement, and then show that this leads to (i.e. entails) a contradiction. Which of the following statements constitutes a contradiction (that is, which of the following evaluates to false)? Suppose A and B to be sets of natural numbers, and to be a natural numbers. Select all that apply. A. O (x E A) ^ (x ¢ B) ^ (x € (AN B)) В. О (хЕ В) — ((х E A) + (х € В))^ (х€ A) C. O (x E A) A (x € B) ^ (AN B = 0) D. O (x E A) ^ (x E B) ^ (x ¢ (AU B)) E. O None of the above
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