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Kőnig's theorem (set theory) - Quiz
  • 1. Which set theory framework is assumed for the proof of Kőnig's theorem provided?
A) Peano arithmetic
B) Zermelo–Fraenkel set theory without choice
C) Zermelo–Fraenkel set theory including the axiom of choice
D) Intuitionistic set theory
  • 2. What is the requirement for the index set I in the theorem?
A) There is no specific constraint given for I
B) I must be a countable set
C) I must be a finite set
D) I must be a well-ordered set
  • 3. What is the condition provided for all i in I regarding A_i and B_i?
A) A_i ≤ B_i
B) A_i = B_i
C) A_i > B_i
D) A_i < B_i
  • 4. What does the axiom of choice imply regarding the condition A < B?
A) There is a bijection between A and B
B) A is a subset of B
C) There is no function from A onto B and B is nonempty
D) B is a subset of A
  • 5. How is the non-emptiness of the product of B_i proven?
A) By definition of the summation
B) By assuming the product is infinite
C) Using the axiom of choice and the fact that factors are nonempty
D) By proving the index set is finite
  • 6. What does the proof show about any function f from the disjoint union of A_i to the product of B_i?
A) That the function is constant
B) That the function is bijective
C) That the function is injective
D) That the function is not surjective
  • 7. For each i, what element is chosen in the proof?
A) An element a_i in A_i
B) The product of all B_i
C) An element b_i in B_i not in the image of A_i
D) The union of all A_i
  • 8. The image of A_i is obtained through the composition of f with what?
A) The identity function
B) The summation over I
C) The projection to B_i
D) The product over I
  • 9. Why is the product of elements b_i not in the image of f?
A) Because the function f is surjective
B) Because each component does not match the image of f
C) Because A_i is empty
D) Because the product is empty
  • 10. What conclusion is drawn from the fact that the product of b_i is not in the image of f?
A) f is a constant function
B) f is injective
C) f does not map the disjoint union of A_i onto the product of B_i
D) the product is equal to the sum
  • 11. For a cardinal κ ≥ ℵ₀, what is the relationship between κ and κcf(κ)?
A) κ > κcf(κ)
B) κ = κcf(κ)
C) κ < κcf(κ)
D) κ ≤ κcf(κ)
  • 12. If κ is regular, the inequality κ < κcf(κ) follows from which theorem?
A) Zorn's lemma
B) Easton's theorem
C) Cantor's theorem
D) Well-ordering theorem
  • 13. What is true about the sum of cardinals approaching a singular cardinal κ?
A) The sum is undefined
B) The sum is less than κ
C) The sum equals κ
D) The sum is greater than κ
  • 14. What theorem provides the only nontrivial constraint on the continuum function for regular cardinals?
A) Cantor's theorem
B) Easton's theorem
C) Schröder-Bernstein theorem
D) Kőnig's theorem
  • 15. For κ ≥ ℵ₀ and λ ≥ 2, what is the relationship between κ and cf(λ^κ)?
A) κ > cf(λ^κ)
B) κ ≤ cf(λ^κ)
C) κ < cf(λ^κ)
D) κ = cf(λ^κ)
  • 16. In the proof of κ < cf(λ^κ), what contradiction is reached if κ ≥ cf(μ) (where μ = λ^κ)?
A) μ > μ
B) μ < μ
C) μ is empty
D) μ = 0
  • 17. What mathematical relationship does Kőnig's theorem imply for any cardinal κ?
A) κ > 2^κ
B) κ < 2^κ
C) κ + 1 = 2^κ
D) κ = 2^κ
  • 18. Was Cantor's theorem proved before or after Kőnig's theorem?
A) At the same time
B) Much earlier
C) It remains unproven
D) Much later
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