- 1. What is the name of the ternary relation defined on a set X?
A) Equivalence relation
B) Linear order
C) Cyclic order
D) Partial order
- 2. Which axiom states that if [a, b, c] holds, then [b, c, a] must also hold?
A) Asymmetry
B) Cyclicity
C) Transitivity
D) Connectedness
- 3. If [a, b, c] implies that [c, b, a] is false, which axiom is satisfied?
A) Asymmetry
B) Cyclicity
C) Connectedness
D) Transitivity
- 4. Which axiom requires that for three distinct elements, either [a, b, c] or [c, b, a] is true?
A) Connectedness
B) Cyclicity
C) Transitivity
D) Asymmetry
- 5. What is a ternary relation called if it satisfies the first three axioms but not necessarily connectedness?
A) Simple cyclic order
B) Linear cyclic order
C) Total cyclic order
D) Partial cyclic order
- 6. Who considered alternative lists of axioms for cyclic orders in 1916 and 1924?
A) Bertrand Russell
B) Kurt Gödel
C) George Cantor
D) Edward Huntington
- 7. Which condition is necessary for [a, b, c] in a linear order induced cyclic order?
A) a < c < b
B) c < b < a
C) a < b < c or b < c < a or c < a < b
D) a < b < c
- 8. What can be done to a deck of cards to illustrate cyclic rearrangements?
A) Cutting
B) Stacking
C) Shuffling
D) Dealing
- 9. What remains after cutting a single point out of a cyclically ordered set?
A) A cyclic order
B) A linear order
C) A partial order
D) A set with no order
- 10. Adjoining 'a' as a greatest element to the linear order results in which type of cut?
A) An asymmetric cut
B) A cyclic rearrangement
C) A principal cut with least element a
D) A cut with greatest element a
- 11. Transitivity in a cyclic order implies that if [a, b, c] and [a, c, d] are true, then which must be true?
A) [c, a, b]
B) [b, c, d]
C) [a, b, d]
D) [d, c, b]
- 12. The axioms for cyclic order are named by analogy with axioms for which type of relation?
A) Partial order
B) Strict linear order
C) Binary equivalence
D) Reflexive relation
- 13. What mathematical object is mentioned as an example of an infinite cycle?
A) A Euclidean plane
B) The set of integers
C) The unit circle, S1
D) A finite sequence
- 14. Why can we not rely on an immediate successor relation in an infinite cyclic order?
A) Successors are only defined for finite sets
B) The set is too small
C) The set is not ordered
D) Points may not have successors
- 15. In the context of the unit circle, what happens if you try to pick a 'next point' after a given point?
A) The next point is defined by a binary relation
B) There is no next point
C) The next point is the origin
D) The next point is always clockwise
- 16. Why can we not use a binary relation to determine which of two points comes first on a circle?
A) Each point follows the other
B) Binary relations are not allowed on infinite sets
C) Only ternary relations exist for circles
D) Points are not distinct
- 17. What type of relation is used to denote that elements occur after each other on a circle?
A) Unary relation
B) Binary relation
C) Ternary relation
D) Quaternary relation
- 18. Which requirement, when dropped from the definition of a cyclic order, results in a partial cyclic order?
A) Cyclic
B) Connected
C) Transitive
D) Asymmetric
- 19. How many notes are in the chromatic scale, which is an example of a finite cycle?
A) Seven
B) Four
C) Three
D) Twelve
- 20. What is a set with a cyclic order called?
A) A linear set
B) A transversal set
C) A binary set
D) A cyclically ordered set
- 21. What is the equivalent of a cyclic order on a set X with n elements?
A) A binary relation
B) A linear permutation
C) A subset
D) A circular permutation
- 22. How many linear orders induce a given cyclic order on a set of n elements?
A) (n-1)!
B) n!
C) n-1
D) n
- 23. How many possible cyclic orders exist for a set of n elements?
A) n!
B) (n-1)!
C) n
D) 2n
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