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