Question 1 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 f(æ) is a function with a domain and codomain of the natural numbers. Select all that apply. A.O (Va, y e N.f(2) = f(v) ^ (Væ, y e N.f(x) + f(w) B. O (Va, y e N.f(2) $(4)) ^ (3y e N.Væ e N.f(æ) + y) 2, y E C. O (3x, y E N.f(x) = f(y)) ^ (Vy e N.3¤ e N.f(æ) = y) D. O (Vy e N.3a e N.f(x) = y) ^ (3y E N.Væ € N.ƒ(x) #y) E. O None of the above Comments:
Question 1 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 f(æ) is a function with a domain and codomain of the natural numbers. Select all that apply. A.O (Va, y e N.f(2) = f(v) ^ (Væ, y e N.f(x) + f(w) B. O (Va, y e N.f(2) $(4)) ^ (3y e N.Væ e N.f(æ) + y) 2, y E C. O (3x, y E N.f(x) = f(y)) ^ (Vy e N.3¤ e N.f(æ) = y) D. O (Vy e N.3a e N.f(x) = y) ^ (3y E N.Væ € N.ƒ(x) #y) E. O None of the above Comments:
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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Transcribed Image Text:Question 1
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 f(x) is a function with a domain and codomain of the natural numbers. Select all that apply.
A. O (Vz, y e N.f(2) = f(y) ^ (vz, y € N.f(z) + f(4)
А.
B. O (Væ, y E N. f (x) + f(y)) ^ (3y E N.Væ € N.f(x) y)
(,y E N.f(x) =
= f(y)) ^ (Vy E N.Jæ E N.f(x)
= U
D. O (Vy E N.3x € N.f(x) = y)
^ (3y E N.Vx E N.f(x) # y)
E. O None of the above
Comments:
Question 2
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 (x E A) ^ (x € B) ^ (AN B = Ø)
B. O (* E A) A (æ ¢ B) ^ (æ E (AN B))
C. O (x E A) ^ (x e B) ^ (x ¢ (AU B))
D. O
(x € B) → ((x E A) → (x E B)) ) ^ (x E A)
E. O None of the above
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