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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) Zermelo–Fraenkel set theory including the axiom of choice
B) Zermelo–Fraenkel set theory without choice
C) Peano arithmetic
D) Intuitionistic set theory
  • 2. What is the requirement for the index set I in the theorem?
A) I must be a finite set
B) There is no specific constraint given for I
C) I must be a well-ordered set
D) I must be a countable 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) A is a subset of B
B) There is a bijection between A and B
C) B is a subset of A
D) There is no function from A onto B and B is nonempty
  • 5. How is the non-emptiness of the product of B_i proven?
A) By assuming the product is infinite
B) By proving the index set is finite
C) By definition of the summation
D) Using the axiom of choice and the fact that factors are nonempty
  • 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 bijective
B) That the function is not surjective
C) That the function is injective
D) That the function is constant
  • 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 A_i is empty
B) Because the function f is surjective
C) Because each component does not match the image of f
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) the product is equal to the sum
C) f does not map the disjoint union of A_i onto the product of B_i
D) f is injective
  • 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) Well-ordering theorem
C) Easton's theorem
D) Cantor's theorem
  • 13. What is true about the sum of cardinals approaching a singular cardinal κ?
A) The sum is less than κ
B) The sum is greater than κ
C) The sum equals κ
D) The sum is undefined
  • 14. What theorem provides the only nontrivial constraint on the continuum function for regular cardinals?
A) Kőnig's theorem
B) Easton's theorem
C) Schröder-Bernstein theorem
D) Cantor'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 = 0
C) μ < μ
D) μ is empty
  • 17. What mathematical relationship does Kőnig's theorem imply for any cardinal κ?
A) κ > 2^κ
B) κ = 2^κ
C) κ < 2^κ
D) κ + 1 = 2^κ
  • 18. Was Cantor's theorem proved before or after Kőnig's theorem?
A) It remains unproven
B) Much earlier
C) At the same time
D) Much later
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