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