Exercise 4. Prove that if f : A → B and g :B → C are bijective, then (go f) : A→C is bijective.Exercise 5. Prove that if x11 and xn+1V6 + xn, then In <3 for all integers n > 1.Exercise 6. Let ECR such that E is nonempty and bounded. Prove that the set -E = {-x : xEE} satisfies inf(E) = – sup(-E).Exercise 7. If x is a rational number and y is irrational, prove that x +y and - y are irrational.Exercise 8. If x, y ER with x < y, prove that x < tx + (1 – t)y < y for all t E (0, 1).Exercise 9. Prove that if A and B are countable sets, then A x B and AU B are countable.Exercise 10. Is the set of all finite sequences of 0s and 1s countable? Justify your answer.Exercise 11. Is the set of all sequences of Os and ls countable? Justify your answer.

Answered: Exercise 4. Prove that if f : A → B and…

Exercise 4. Prove that if f : A → B and g : B → C are bijective, then (gof) : A→C is bijective. then r 3 for all intogomo Exercise 5, Prove that if T. = 1 and r

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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Let f be a function that takes elements of R?«R? into R? as follows: f(u, v) = |u1v1|+|u2v2|, where u = (u1, U2) e R² and v = (v1, v2) E R². Show that f cannot be an inner product by showing that one of the properties is not satisfied.
2. Determine if the function (u, v) = U₁V₂ + U₂V1, where u = each axiom to determine if it holds or fails. Show all work. (u₁, ₂) and 7 = (V₁, V₂) defines an inner product on R². Check
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