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Proving theorems and taking stocks.
Exercise 3-30 | Elements of Set Theory
30. Assume that F:𝒫A→𝒫A and that F has the monotonicity property:
X ⊆ Y ⊆ A ⇒ F(X) ⊆ F(Y).
Define B = ⋂{X ⊆ A | F(X) ⊆ X} and C = ⋃{X ⊆ A | X ⊆ F(X)}.
(a) Show that F(B) = B and F(C) = C.
(b) Show that if F(X) = X, then B ⊆ X ⊆ C.
X ⊆ Y ⊆ A ⇒ F(X) ⊆ F(Y).
Define B = ⋂{X ⊆ A | F(X) ⊆ X} and C = ⋃{X ⊆ A | X ⊆ F(X)}.
(a) Show that F(B) = B and F(C) = C.
(b) Show that if F(X) = X, then B ⊆ X ⊆ C.
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Видео
Exercise 3-29 | Elements of Set Theory
Просмотров 122 месяца назад
29. Assume that f:A→B and define a function G:B→𝒫A by G(b) = {x ∈ A | f(x) = b}. Show that if f maps A onto B, then G is one-to-one. Does the converse hold?
Exercise 3-28 | Elements of Set Theory
Просмотров 92 месяца назад
28. Assume that f is a one-to-one function from A into B, and that G is the function with dom G = 𝒫A defined by the equation G(X) = f⟦X⟧. Show that G maps 𝒫A one-to-one into 𝒫B.
Exercise 3-27 | Elements of Set Theory
Просмотров 92 месяца назад
27. Show that dom(F∘G) = G⁻¹⟦dom F⟧ for any sets F and G. (F and G need not be functions.)
Exercise 3-26 | Elements of Set Theory
Просмотров 72 месяца назад
26. Prove the second halves of parts (a) and (b) of Theorem 3K.
Exercise 3-25 | Elements of Set Theory
Просмотров 92 месяца назад
25. (a) Assume that G is a one-to-one function. Show that G∘G⁻¹ is I_{ran G}, the identity function on ran G. (b) Show that the result of part (a) holds for any function G, not necessarily one-to-one.
Exercise 3-24 | Elements of Set Theory
Просмотров 132 месяца назад
24. Show that for a function F, F⁻¹⟦A⟧ = {x ∈ dom F | F(x) ∈ A}.
Exercise 3-23 | Elements of Set Theory
Просмотров 1152 месяца назад
23. Let I_A be the identity function on the set A. Show that for any sets B and C, B∘I_A = B↾A and I_A⟦C⟧ = A∩C.
Exercise 3-22 | Elements of Set Theory
Просмотров 752 месяца назад
22. Show that the following are correct for any sets. (a) A ⊆ B ⇒ F⟦A⟧ ⊆ F⟦B⟧. (b) (F∘G)⟦A⟧ = F⟦G⟦A⟧⟧. (c) Q↾(A ∪ B) = (Q↾A) ∪ (Q↾B)
Exercise 3-21 | Elements of Set Theory
Просмотров 1352 месяца назад
21. Show that R∘(S∘T) = (R∘S)∘T for any sets R, S, and T.
Exercise 3-20 | Elements of Set Theory
Просмотров 152 месяца назад
20. Show that F↾A = F ∩ (A ⨯ ran F).
Exercise 3-19 | Elements of Set Theory
Просмотров 192 месяца назад
19. Let A = {⟨∅, {∅, {∅}}⟩, ⟨{∅}, ∅⟩}. Evaluate each of the following: A(∅), A⟦∅⟧, A⟦{∅}⟧, A⟦{∅, {∅}}⟧, A⁻¹, A∘A, A↾∅, A↾{∅}, A↾{∅, {∅}}, ⋃⋃A.
Exercise 3-18 | Elements of Set Theory
Просмотров 132 месяца назад
Let R be the set {⟨0, 1⟩, ⟨0, 2⟩, ⟨0, 3⟩, ⟨1, 2⟩, ⟨1, 3⟩, ⟨2, 3⟩}. Evaluate the following R∘R, R↾{1}, R⁻¹↾{1}, R⟦{1}⟧, and R⁻¹⟦{1}⟧.
Exercise 3-17 | Elements of Set Theory
Просмотров 122 месяца назад
17. Show that the composition of two single-rooted sets is again single-rooted. Conclude that the composition of two one-to-one functions is again one-to-one.
Exercise 3-16 | Elements of Set Theory
Просмотров 102 месяца назад
16. Show that there is no set to which every function belongs.
Exercise 3-15 | Elements of Set Theory
Просмотров 722 месяца назад
Exercise 3-15 | Elements of Set Theory
Exercise 3-14 | Elements of Set Theory
Просмотров 143 месяца назад
Exercise 3-14 | Elements of Set Theory
Exercise 3-13 | Elements of Set Theory
Просмотров 273 месяца назад
Exercise 3-13 | Elements of Set Theory
Exercise 3-12 | Elements of Set Theory
Просмотров 143 месяца назад
Exercise 3-12 | Elements of Set Theory
Exercise 3-11 | Elements of Set Theory
Просмотров 163 месяца назад
Exercise 3-11 | Elements of Set Theory
Exercise 3-10 | Elements of Set Theory
Просмотров 143 месяца назад
Exercise 3-10 | Elements of Set Theory
Exercise 3-9 | Elements of Set Theory
Просмотров 133 месяца назад
Exercise 3-9 | Elements of Set Theory
Exercise 3-8 | Elements of Set Theory
Просмотров 73 месяца назад
Exercise 3-8 | Elements of Set Theory
Exercise 3-7 | Elements of Set Theory
Просмотров 153 месяца назад
Exercise 3-7 | Elements of Set Theory
Exercise 3-6 | Elements of Set Theory
Просмотров 133 месяца назад
Exercise 3-6 | Elements of Set Theory
Exercise 3-5 | Elements of Set Theory
Просмотров 183 месяца назад
Exercise 3-5 | Elements of Set Theory
Exercise 3-4 | Elements of Set Theory
Просмотров 133 месяца назад
Exercise 3-4 | Elements of Set Theory
Exercise 3-3 | Elements of Set Theory
Просмотров 103 месяца назад
Exercise 3-3 | Elements of Set Theory
Exercise 3-2 | Elements of Set Theory
Просмотров 93 месяца назад
Exercise 3-2 | Elements of Set Theory
Exercise 3-1 | Elements of Set Theory
Просмотров 243 месяца назад
Exercise 3-1 | Elements of Set Theory
Hello there! Found out about your project a few weeks ago. Very cool. I love your work. I hope you have the energy/motivation to finish the entire book's exercises. The content of your videos, to my eye, look like professional mathematics. Best wishes, I hope for your success and I hope you keep doing mathematics (also) on RUclips.
What a great project, thanks for sharing it. Greetings.
Thank you! Just trying to share my love for math :)
What are those funny symbols
Which ones?
Needed the hint, but then this was a fun one!
Glad you enjoyed it!
Good 👍🏻