3.7.21. Write the negation of the following statements without using the negation symbol -. Also, for each statement, decide whether it is true or false. Explain your answer briefly. (a) (Vr E R)(3y E R)(r² > y²) (b) (3r e Z) [(x² = (x + 1)*) → (r* € Z)] ©2017 Shay Fuchs. All rights reserved. 79 3.7. EXERCISES FOR CHAPTER 3 CHAPTER 3. INFORMAL LOGIC (c) (Vn e N)[(n – 1)* + n° + (n + 1)*] (d) [(V ER)(x > 0)) = [(Vx € R)(x = 1+1)} (e) (Vz E R)( (z2 < -1) = (x + 1)2 = x2+ 1]]
3.7.21. Write the negation of the following statements without using the negation symbol -. Also, for each statement, decide whether it is true or false. Explain your answer briefly. (a) (Vr E R)(3y E R)(r² > y²) (b) (3r e Z) [(x² = (x + 1)*) → (r* € Z)] ©2017 Shay Fuchs. All rights reserved. 79 3.7. EXERCISES FOR CHAPTER 3 CHAPTER 3. INFORMAL LOGIC (c) (Vn e N)[(n – 1)* + n° + (n + 1)*] (d) [(V ER)(x > 0)) = [(Vx € R)(x = 1+1)} (e) (Vz E R)( (z2 < -1) = (x + 1)2 = x2+ 1]]
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter7: Real And Complex Numbers
Section7.1: The Field Of Real Numbers
Problem 20E: Give counterexamples for the following statements.
If and are irrational, then is irrational.
If ...
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Question
3.7.21
Parts a)b)e
![(b) For any positive real number r, there is a natural number n, for which
< r.
3.7.21. Write the negation of the following statements without using the negation symbol .
Also, for each statement, decide whether it is true or false. Explain your answer briefly.
(a) (Vr E R)(3y e R)(r² > y?)
(b) (3r € Z) [(x² = (r + 1)*) → (r* € Z)]
©2017 Shay Fuchs. All rights reserved.
79
3.7. EXERCISES FOR CHAPTER 3
CHAPTER 3. INFORMAL LOGIC
(c) (Vn e N)(n– 1)* + n° # (n + 1)°]
(d) [(V ER)(x> 0) = [(Vr € R)(x = x+1)]
(e) (Vz ER)((z < -1) = [(x + 1)² = x² + 1]]
(f) (Vz ER)(r> 0) (3n E N)(n-z> 1)]
(g) (V ER)(3y E R)(r+y)? + v°)
(h) (3y ER)(Va E R)(a+ yl |r|+lvl)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb149039b-c609-4874-85d1-64f29a0551b7%2F149245ed-4689-44b4-afea-652453cf1f8c%2Fuegcpjm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(b) For any positive real number r, there is a natural number n, for which
< r.
3.7.21. Write the negation of the following statements without using the negation symbol .
Also, for each statement, decide whether it is true or false. Explain your answer briefly.
(a) (Vr E R)(3y e R)(r² > y?)
(b) (3r € Z) [(x² = (r + 1)*) → (r* € Z)]
©2017 Shay Fuchs. All rights reserved.
79
3.7. EXERCISES FOR CHAPTER 3
CHAPTER 3. INFORMAL LOGIC
(c) (Vn e N)(n– 1)* + n° # (n + 1)°]
(d) [(V ER)(x> 0) = [(Vr € R)(x = x+1)]
(e) (Vz ER)((z < -1) = [(x + 1)² = x² + 1]]
(f) (Vz ER)(r> 0) (3n E N)(n-z> 1)]
(g) (V ER)(3y E R)(r+y)? + v°)
(h) (3y ER)(Va E R)(a+ yl |r|+lvl)
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