By Smith D., Eggen M., Andre R.

ISBN-10: 0495562025

ISBN-13: 9780495562023

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We may assume this hypothesis since it is given in the statement of the theorem. Suppose x is odd. We assume that the antecedent P is true. The goal is to derive the consequent Q as our last step. From the definition of odd, x = 2k + 1 for some integer k. This deduction is the replacement Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. ” We now have an equation to use. Then x + 1 = (2k + 1) + 1 for some integer k. This is another ގ.

A) (∀x)(Ey)(x + y = 0). (b) (Ex)(∀y)(x + y = 0). (c) (Ex)(Ey)(x2 + y2 = −1). (d) (∀x)[x > 0 ⇒ ( Ey)(y < 0 ∧ xy > 0)]. (e) (∀y)(Ex)(∀z)(xy = xz). (f) (Ex)(∀y)(x ≤ y). (g) (∀y)(Ex)(x ≤ y). y)(y < 0 ∧ y + 3 > 0). x)(∀y)(x = y2). x)(x = y2). y)(∀w)(w2 > x − y). Let A(x) be an open sentence with variable x. 2 (a). 2 (a) is false. 2 (b). x) A (x) is equivalent to (Ex)[A(x) ∧ (∀y)(A(y) ⇒ x = y)]. x) A (x). ૺ 12. ” (b) Write the symbolic form of the statement of the Mean Value Theorem. ” x→a (d) Write a useful denial of each sentence in parts (a), (b), and (c).

That is, every real number is positive, zero or negative. (∀x)(x ≥ 3) is false because there are (many) real numbers x for which x ≥ 3 is false. (∀x)(|x| > 0) is false, because 0 is not in the truth set. There are many ways to express a quantified sentence in English. Look for key words such as “for all,” “for every,” “for each,” or similar words that require universal quantifiers. Look for phrases such as “some,” “at least one,” “there exist(s),” “there is (are),” and others that indicate existential quantifiers.

### A transition to advanced mathematics by Smith D., Eggen M., Andre R.

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