1Which three properties define a partial order relation on a set?
introduction
Easy
A.Irreflexive, symmetric, and transitive
B.Irreflexive, antisymmetric, and asymmetric
C.Reflexive, antisymmetric, and transitive
D.Reflexive, symmetric, and transitive
Correct Answer: Reflexive, antisymmetric, and transitive
Explanation:
A relation is called a partial order if it is reflexive, antisymmetric, and transitive. This forms the foundation of a partially ordered set (poset).
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2In an ordered set, the antisymmetric property states that if and , then which of the following must be true?
ordered sets
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
The antisymmetric property ensures that two distinct elements cannot both precede each other. If and , they must be the same element, so .
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3What is a totally ordered set (also known as a chain)?
ordered sets
Easy
A.A poset with a finite number of elements
B.A poset where no elements are comparable
C.A poset that has a maximum and a minimum element
D.A poset where every two elements are comparable
Correct Answer: A poset where every two elements are comparable
Explanation:
A totally ordered set is a partially ordered set in which any two elements and are comparable, meaning either or .
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4For a relation on a set to be reflexive, which condition must hold for all ?
ordered sets
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
The reflexive property means every element is related to itself, so must hold for all in the set .
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5In a Hasse diagram, which types of edges are intentionally omitted to simplify the drawing?
Hasse diagrams of partial ordered sets
Easy
A.Edges connecting to the minimal elements
B.Symmetric and asymmetric edges
C.Reflexive loops and transitive edges
D.Edges between comparable elements
Correct Answer: Reflexive loops and transitive edges
Explanation:
Hasse diagrams omit reflexive self-loops and edges implied by transitivity to create a cleaner and simpler representation of a partial order.
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6How is the relation visually represented in a standard Hasse diagram?
Hasse diagrams of partial ordered sets
Easy
A.The node for is placed higher than the node for , with a sequence of upward edges connecting them.
B.The node for is placed higher than the node for , with a sequence of downward edges connecting them.
C.The nodes and are placed on the same horizontal level.
D.A directed arrow points from to .
Correct Answer: The node for is placed higher than the node for , with a sequence of upward edges connecting them.
Explanation:
In a Hasse diagram, larger elements are placed higher up. If , will appear above and can be reached by tracing edges upwards.
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7What is another common name in computer science for finding a consistent enumeration of a finite poset?
consistent enumeration
Easy
A.Binary search
B.Breadth-first search
C.Hashing
D.Topological sorting
Correct Answer: Topological sorting
Explanation:
A consistent enumeration of a poset assigns an integer to each element such that the order is preserved. This is equivalent to topological sorting of a Directed Acyclic Graph (DAG).
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8If a function is a consistent enumeration for a poset , what must be true if in ?
consistent enumeration
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
A consistent enumeration maps elements of a poset to numbers such that the order is preserved. Thus, if strictly precedes , its mapped value must be strictly less than .
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9What is the supremum of a subset in a partially ordered set?
supremum and infimum
Easy
A.The maximum element in the entire poset
B.Any upper bound of
C.The least upper bound of
D.The greatest lower bound of
Correct Answer: The least upper bound of
Explanation:
The supremum of a set is defined as its least upper bound (LUB). It is an upper bound of that is smaller than or equal to all other upper bounds of .
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10What is the infimum of a subset in a partially ordered set?
supremum and infimum
Easy
A.The minimal element of the entire poset
B.The greatest lower bound of
C.Any lower bound of
D.The least upper bound of
Correct Answer: The greatest lower bound of
Explanation:
The infimum of a set is defined as its greatest lower bound (GLB). It is a lower bound of that is greater than or equal to all other lower bounds of .
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11If a subset of a poset has a supremum, how many suprema can it possibly have?
supremum and infimum
Easy
A.Exactly one
B.At least two
C.Depends on the size of the set
D.Infinitely many
Correct Answer: Exactly one
Explanation:
If a least upper bound (supremum) exists for a subset in a poset, it is unique due to the antisymmetric property of partial orders.
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12Two posets and are isomorphic if there exists a bijection that satisfies which of the following?
isomorphic (similar) ordered sets
Easy
A. if and only if
B. if and only if
C. if and only if
D. if and only if
Correct Answer: if and only if
Explanation:
An order isomorphism is a bijective function between two posets that perfectly preserves the order relation in both directions.
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13A set is called well-ordered if every non-empty subset contains which of the following?
well-ordered sets
Easy
A.A least element
B.An upper bound
C.A greatest element
D.Both a least and a greatest element
Correct Answer: A least element
Explanation:
By definition, a well-ordered set is a totally ordered set in which every non-empty subset has a least (minimum) element.
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14Which of the following standard mathematical sets is a well-ordered set under the usual "less than or equal to" relation?
well-ordered sets
Easy
A.The set of natural numbers
B.The set of real numbers
C.The set of rational numbers
D.The set of all integers
Correct Answer: The set of natural numbers
Explanation:
The set of natural numbers is well-ordered because any non-empty subset of natural numbers has a smallest element. Integers, rationals, and reals do not generally have this property (e.g., negative integers have no minimum).
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15By definition, a lattice is a partially ordered set in which every pair of elements has which of the following?
lattices and bounded lattices
Easy
A.Only an infimum
B.Both a supremum and an infimum
C.Only a supremum
D.A complement
Correct Answer: Both a supremum and an infimum
Explanation:
A lattice is a poset where any two elements and have a unique least upper bound (supremum/join) and a unique greatest lower bound (infimum/meet).
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16In a bounded lattice, what do the special symbols $0$ and $1$ generally represent?
lattices and bounded lattices
Easy
A.The global maximum and global minimum elements, respectively
B.The global minimum and global maximum elements, respectively
C.The first and second elements in a chain
D.The least upper bound and greatest lower bound of a specific subset
Correct Answer: The global minimum and global maximum elements, respectively
Explanation:
A bounded lattice has a unique greatest element (often denoted as $1$ or ) and a unique least element (often denoted as $0$ or ).
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17A lattice is considered distributive if which two operations distribute over each other?
distributive lattices
Easy
A.Union () and Complement ()
B.Meet () and Join ()
C.Supremum and Minimum
D.Addition () and Multiplication ()
Correct Answer: Meet () and Join ()
Explanation:
In a distributive lattice, the meet operation distributes over the join operation: , and vice versa.
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18Which of the following is a classic example of a distributive lattice?
distributive lattices
Easy
A.The lattice of all subgroups of a group
B.The power set of a set under subset inclusion
C.A diamond lattice ()
D.A pentagon lattice ()
Correct Answer: The power set of a set under subset inclusion
Explanation:
The power set of a set, ordered by inclusion, forms a distributive lattice where the meet is set intersection and the join is set union. and are the standard examples of non-distributive lattices.
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19In a bounded lattice with minimum $0$ and maximum $1$, an element is a complement of if which of the following holds?
complements and complemented lattices
Easy
A. and
B. and
C. and
D. and
Correct Answer: and
Explanation:
By definition, an element is a complement of if their join (least upper bound) is the global maximum ($1$) and their meet (greatest lower bound) is the global minimum ($0$).
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20What is a complemented lattice?
complements and complemented lattices
Easy
A.A bounded lattice where every element has at least one complement
B.A lattice that contains no bounds $0$ or $1$
C.A lattice where every element has exactly two complements
D.A lattice where only the bounds $0$ and $1$ have complements
Correct Answer: A bounded lattice where every element has at least one complement
Explanation:
A complemented lattice is defined as a bounded lattice (having a maximum $1$ and minimum $0$) in which every element has at least one complement.
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21Let be ordered by divisibility. Which of the following pairs of elements is incomparable?
ordered sets
Medium
A.2 and 8
B.2 and 6
C.4 and 6
D.3 and 12
Correct Answer: 4 and 6
Explanation:
In a poset ordered by divisibility, if and only if divides . Because 4 does not divide 6, and 6 does not divide 4, these two elements are incomparable.
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22In the Hasse diagram of the poset ordered by divisibility, which element is an immediate predecessor of 12?
Hasse diagrams of partial ordered sets
Medium
A.8
B.3
C.2
D.4
Correct Answer: 4
Explanation:
An immediate predecessor of means divides , and there is no intermediate element in the set such that and . The divisors of 12 in the set are 1, 2, 3, 4, 6. The immediate predecessors of 12 are 4 and 6. Therefore, 4 is the correct choice.
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23A consistent enumeration (topological sort) of a poset is a linear ordering such that:
consistent enumeration
Medium
A.The maximal element in the poset is always first in the linear order.
B.All incomparable elements in the poset become equal in the linear order.
C.If in the poset, then in the linear order.
D.If in the poset, then in the linear order.
Correct Answer: If in the poset, then in the linear order.
Explanation:
A consistent enumeration extends a partial order to a total order while strictly preserving all original relationships. Thus, if precedes in the partial order, it must also precede in the linear order.
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24Let be the poset of the power set of ordered by set inclusion . What is the supremum of and ?
supremum and infimum
Medium
A.{c}
B.{a, b, c}
C.{a, b}
D.
Correct Answer: {a, b}
Explanation:
In a power set lattice ordered by inclusion, the supremum (least upper bound) of two sets is their union. Therefore, .
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25Consider the poset ordered by divisibility. What is the infimum of the subset in ?
supremum and infimum
Medium
A.6
B.1
C.
D.It does not exist
Correct Answer: It does not exist
Explanation:
The infimum is the greatest lower bound. In the set , there are no elements that divide both 2 and 3 (the number 1 is not in the set). Therefore, there is no lower bound at all, meaning the infimum does not exist.
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26Two posets and are isomorphic if there exists a bijection such that for all :
isomorphic (similar) ordered sets
Medium
A. if and only if
B. implies
C.they have the exact same number of elements without structural constraints
D. implies (but the converse is not required)
Correct Answer: if and only if
Explanation:
An order isomorphism requires a bijective function that perfectly preserves the order structure in both directions (an if-and-only-if condition).
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27Which of the following ordered sets is a well-ordered set under the standard less-than-or-equal-to relation ()?
well-ordered sets
Medium
A.The open interval of real numbers
B.The set of all integers
C.The set of all natural numbers
D.The set of all positive rational numbers
Correct Answer: The set of all natural numbers
Explanation:
A set is well-ordered if every non-empty subset has a least element. satisfies this property, whereas , , and have non-empty subsets without a least element.
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28Let be a lattice. Which of the following identifies the absorption laws in ?
lattices and bounded lattices
Medium
A.
B. and
C. and
D. and
Correct Answer: and
Explanation:
The absorption laws state that an element combined with a meet or join of itself and another element results in the original element.
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29Which of the following lattices is NEVER a distributive lattice?
distributive lattices
Medium
A.The set of natural numbers under the standard relation
B.A totally ordered set (chain) with 4 elements
C.The pentagon lattice
D.The power set of a set under inclusion
Correct Answer: The pentagon lattice
Explanation:
A fundamental theorem states that a lattice is distributive if and only if it does not contain a sublattice isomorphic to the pentagon lattice or the diamond lattice .
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30In a bounded lattice , an element is a complement of an element if:
complements and complemented lattices
Medium
A. and
B. and
C. and
D. and
Correct Answer: and
Explanation:
By definition, the complement of is an element such that their supremum is the greatest element (1) and their infimum is the least element (0).
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31Let be the set of all positive divisors of 30, ordered by divisibility. What are the bounds $0$ (least element) and $1$ (greatest element) of this lattice?
lattices and bounded lattices
Medium
A.,
B.,
C.,
D.,
Correct Answer: ,
Explanation:
In a lattice of divisors of ordered by divisibility, the least element is 1 (since 1 divides everything) and the greatest element is (since everything divides ). Thus, and .
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32When drawing a Hasse diagram for a poset, which of the following relations are NOT represented by an edge?
Hasse diagrams of partial ordered sets
Medium
A.Reflexivity relations ()
B.Both reflexivity and transitivity relations
C.Immediate predecessors
D.Transitivity relations where a middle element exists ( and )
Correct Answer: Both reflexivity and transitivity relations
Explanation:
In a Hasse diagram, loops representing reflexivity are omitted, and edges representing transitivity (when an intermediate element exists) are also omitted to simplify the diagram. Only covering relations (immediate predecessors) are drawn.
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33A relation on a set is a partial order if it is:
ordered sets
Medium
A.Irreflexive, antisymmetric, and transitive
B.Reflexive, symmetric, and transitive
C.Reflexive, antisymmetric, and transitive
D.Reflexive, antisymmetric, and symmetric
Correct Answer: Reflexive, antisymmetric, and transitive
Explanation:
A partial order is defined by three properties: reflexive (), antisymmetric (if and , then ), and transitive (if and , then ).
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34Let . Which of the following relations on is NOT a partial order?
introduction
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
This relation contains both and but . This violates the antisymmetry property, so it is not a partial order.
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35Suppose a project has tasks . The prerequisites form a poset where , , and . Which of the following is a valid consistent enumeration of these tasks?
consistent enumeration
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
A consistent enumeration requires that if , appears before . must precede and , and must precede . The sequence satisfies all these conditions.
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36Let be a distributive lattice. If and , what can be definitively concluded about and ?
distributive lattices
Medium
A. is the complement of
B.
C.
D.No conclusion can be drawn.
Correct Answer:
Explanation:
In a distributive lattice, an element is uniquely determined by its join and meet with another element. This cancellation property guarantees that .
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37Consider the lattice under divisibility. What is the complement of 2?
complements and complemented lattices
Medium
A.15
B.30
C.3
D.5
Correct Answer: 15
Explanation:
In , the least element is 1 and the greatest is 30. The complement of 2 is an element such that (infimum) and (supremum). The only element satisfying both is 15.
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38Let be a total order and be a poset. Are and isomorphic?
isomorphic (similar) ordered sets
Medium
A.Yes, because both are bounded lattices.
B.Yes, because there is a bijection between them.
C.No, because they have a different number of elements.
D.No, because is not a total order.
Correct Answer: No, because they have a different number of elements.
Explanation:
has 3 elements, whereas has elements (the empty set and ). An isomorphism requires a bijection, which is impossible between sets of different sizes.
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39If is a well-ordered set, which of the following MUST be true?
well-ordered sets
Medium
A. is a totally ordered set (chain).
B.Both A and C are true.
C. contains a maximum element.
D. has no infinite descending chains.
Correct Answer: Both A and C are true.
Explanation:
A well-ordered set is always totally ordered, and the well-ordering principle (every non-empty subset has a least element) prevents the existence of any infinite descending chains.
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40In a lattice, the operations (supremum) and (infimum) satisfy the idempotent laws. What is the idempotent law for ?
supremum and infimum
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The idempotent law dictates that an operation applied to an element and itself yields the element itself. For the supremum, this is written as .
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41How many non-isomorphic partial orders (up to isomorphism) can be defined on a set of exactly 3 elements?
ordered sets
Hard
A.5
B.3
C.6
D.4
Correct Answer: 5
Explanation:
On a 3-element set, the non-isomorphic posets are: (1) the antichain (3 isolated nodes), (2) a chain of 2 nodes and 1 isolated node, (3) a 'V' shape (1 minimal, 2 maximal elements), (4) a '' shape (2 minimal, 1 maximal elements), and (5) a totally ordered chain of 3 nodes. Thus, there are exactly 5.
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42Consider the Boolean lattice , represented by the subsets of an -element set ordered by inclusion. What is the minimum for which the Hasse diagram of is non-planar?
Hasse diagrams of partial ordered sets
Hard
A.4
B.5
C.3
D.6
Correct Answer: 4
Explanation:
The Hasse diagram of is a line, is a square, and is a cube, all of which are planar graphs. However, is a tesseract graph, which contains a subdivision of the non-planar bipartite graph . Therefore, is the minimum.
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43Let be a poset with elements and cover relations , , and . What is the total number of valid consistent enumerations (linear extensions) of ?
consistent enumeration
Hard
A.6
B.4
C.8
D.5
Correct Answer: 5
Explanation:
The poset has an 'N' shape. The minimal elements are and . If is chosen first, the remaining elements must satisfy , yielding 3 permutations: . If is chosen first, must be chosen next (as it's the only remaining minimal element), leaving which gives 2 permutations. Total = linear extensions.
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44Let be the set of all continuous real-valued functions on . For , define a partial order where if for all . Which of the following statements is true regarding ?
supremum and infimum
Hard
A.It forms a distributive lattice.
B.It does not form a lattice because it is not well-ordered.
C.It forms a lattice, but it is not distributive.
D.It does not form a lattice because the supremum of two continuous functions is not guaranteed to be continuous.
Correct Answer: It forms a distributive lattice.
Explanation:
For any two continuous functions and , their supremum is and infimum is . Both maximum and minimum of two continuous functions are continuous, so is a lattice. Furthermore, the pointwise max/min operations distribute over each other, making it a distributive lattice.
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45Which of the following conditions is both necessary and sufficient for two finite partially ordered sets and to be isomorphic?
isomorphic (similar) ordered sets
Hard
A.They have the exact same number of minimal and maximal elements and the same height.
B.There exists a bijective function such that if , then .
C.There exists a bijective function such that .
D.Their Hasse diagrams contain the same number of vertices and edges.
Correct Answer: There exists a bijective function such that .
Explanation:
An isomorphism between posets requires a bijection that preserves the order in both directions. The mere existence of an order-preserving bijection (Option D) is insufficient because its inverse might not be order-preserving.
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46Let and be two well-ordered sets. Under which of the following Cartesian product operations is the resulting set NOT necessarily well-ordered?
well-ordered sets
Hard
A.The Lexicographical order on
B.The Lexicographical order on
C.The Product order on
D.The resulting sets are well-ordered in all the above cases.
Correct Answer: The Product order on
Explanation:
The product order defines if and only if and . This typically yields a partial order that is not totally ordered (elements can be incomparable), thus it cannot be well-ordered. Lexicographical orders on well-ordered sets, however, yield well-ordered sets.
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47Consider the lattice , consisting of a minimum element , a maximum element , and mutually incomparable elements strictly between and . For which values of is a modular lattice but NOT a distributive lattice?
lattices and bounded lattices
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
The lattice is the diamond lattice, which is the classic example of a lattice that is modular but non-distributive. All for contain as a sublattice, so they are modular but not distributive. is the Boolean lattice , which is distributive.
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48Birkhoff's Representation Theorem states that any finite distributive lattice is isomorphic to the lattice of lower sets (ideals) of a uniquely determined underlying poset . What are the elements of this underlying poset in relation to ?
distributive lattices
Hard
A.The maximal chains of
B.The join-irreducible elements of
C.The set of all atoms of
D.The meet-irreducible elements of
Correct Answer: The join-irreducible elements of
Explanation:
Birkhoff's Theorem establishes that every finite distributive lattice can be uniquely represented as the lattice of lower sets of the poset constructed specifically from its join-irreducible elements.
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49Let be a bounded, distributive lattice. If an element has a complement , which of the following statements must rigorously hold true?
complements and complemented lattices
Hard
A. is only guaranteed to be unique if is a Boolean algebra.
B. is the unique complement of .
C.Every other element in must also have at least one complement.
D. may not be unique, but any other complement must satisfy .
Correct Answer: is the unique complement of .
Explanation:
A fundamental property of distributive lattices is that if an element possesses a complement, that complement is strictly unique. The entire lattice does not need to be complemented (a Boolean algebra) for this specific uniqueness to hold.
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50Let be a strict partial order relation on a set consisting of elements. What is the absolute maximum number of ordered pairs that can be contained in ?
introduction
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
A strict partial order is irreflexive and transitive, making it asymmetric. The maximum number of edges in a directed acyclic graph that is transitively closed corresponds to a strict total order (a tournament graph). The number of pairs in a strict total order of elements is .
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51Consider the partially ordered set formed by all partitions of an -element set, ordered by refinement (a partition is 'less than' another if it is finer). What is the total number of elements in the longest possible chain in this poset?
ordered sets
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
The finest partition (minimal element) has blocks (singleton sets), and the coarsest partition (maximal element) has 1 block (the whole set). Moving one step up in a maximal chain corresponds to merging exactly two blocks into one, decreasing the block count by 1. Therefore, there are distinct partitions in the longest chain.
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52When drawing the Hasse diagram for the poset of positive divisors of (where are distinct primes) under divisibility, how many vertices and edges will the diagram contain?
Hasse diagrams of partial ordered sets
Hard
A.12 vertices and 24 edges
B.24 vertices and 46 edges
C.24 vertices and 52 edges
D.24 vertices and 36 edges
Correct Answer: 24 vertices and 46 edges
Explanation:
The number of vertices is . The poset is the product of chains of sizes 3, 4, and 2. The formula for the number of edges of a product of chains is . Thus, edges = .
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53Consider a poset whose Hasse diagram forms a directed tree rooted at a single minimum element (an arborescence). The tree has root with children and . has one child . has two children and . What is the total number of consistent enumerations of this poset?
consistent enumeration
Hard
A.120
B.60
C.20
D.36
Correct Answer: 20
Explanation:
The number of linear extensions of a tree-shaped poset rooted at a minimum element is given by , where is the size of the subtree rooted at . The sizes are: A=6, B=2, C=3, D=1, E=1, F=1. The result is .
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54Let be a lattice. An element is called join-irreducible if implies or . In the lattice of positive divisors of under divisibility, how many completely join-irreducible elements are there?
supremum and infimum
Hard
A.5
B.3
C.12
D.4
Correct Answer: 4
Explanation:
In the divisor lattice, join-irreducible elements correspond precisely to elements with exactly one lower cover. In number theory, these are exactly the prime powers. For , the prime powers dividing 60 are , and . Hence, there are exactly 4 join-irreducible elements.
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55Let be a poset, and its dual be the poset with the inverted relation (). A poset is self-dual if . Which of the following finite posets is NOT self-dual?
isomorphic (similar) ordered sets
Hard
A.The pentagon lattice .
B.The lattice of divisors of 36 under divisibility.
C.The Boolean lattice .
D.The poset formed by the positive divisors of 30 strictly greater than 1 under divisibility.
Correct Answer: The poset formed by the positive divisors of 30 strictly greater than 1 under divisibility.
Explanation:
The divisors of form a Boolean lattice . Omitting 1 removes the single minimum element but retains the single maximum element (30). The resulting poset has 3 minimal elements and 1 maximal element. Its dual would have 1 minimal and 3 maximal elements, so they cannot be isomorphic.
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56In the standard foundations of mathematics (Zermelo-Fraenkel set theory), the Well-Ordering Theorem states that every set can be well-ordered. This theorem is rigorously logically equivalent to which of the following statements?
well-ordered sets
Hard
A.The Continuum Hypothesis.
B.The Axiom of Choice.
C.Every partially ordered set has a maximal element.
D.Every totally ordered set is order-isomorphic to a subset of the real numbers.
Correct Answer: The Axiom of Choice.
Explanation:
In ZF set theory, the Well-Ordering Theorem, Zorn's Lemma (which mentions posets having maximal elements, but requires every chain to have an upper bound), and the Axiom of Choice are fundamentally logically equivalent.
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57Consider a bounded finite modular lattice . Which fundamental theorem or condition ensures that all maximal chains between the minimum element and the maximum element contain the exact same number of elements?
lattices and bounded lattices
Hard
A.The Jordan-Dedekind chain condition.
B.The Dedekind-MacNeille Completion Theorem.
C.Dilworth's Theorem.
D.Birkhoff's Representation Theorem.
Correct Answer: The Jordan-Dedekind chain condition.
Explanation:
The Jordan-Dedekind chain condition states that if a finite poset satisfies this condition, all maximal chains between any two comparable elements have the same length. It is a well-known property that all finite modular (and thus distributive) lattices naturally satisfy the Jordan-Dedekind chain condition.
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58Let be a lattice. The Theorem provides a structural characterization of distributive lattices. According to this theorem, is distributive if and only if:
distributive lattices
Hard
A.Every element in has at most one complement.
B. contains no sublattice isomorphic to the Boolean lattice .
C. can be mapped to a linear order.
D. contains no sublattice isomorphic to the pentagon lattice or the diamond lattice .
Correct Answer: contains no sublattice isomorphic to the pentagon lattice or the diamond lattice .
Explanation:
A lattice is distributive if and only if it does not contain a sublattice isomorphic to (the pentagon lattice) or (the diamond lattice). This is the definitive forbidden-sublattice characterization of distributive lattices.
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59A Boolean algebra is fundamentally defined as a lattice that is bounded, distributive, and complemented. How many non-isomorphic Boolean algebras exist that contain exactly 12 elements?
complements and complemented lattices
Hard
A.2
B.1
C.0
D.4
Correct Answer: 0
Explanation:
Any finite Boolean algebra must be isomorphic to the power set of some finite set under inclusion. Therefore, the cardinality of any finite Boolean algebra must be strictly a power of 2 (i.e., ). Since 12 is not a power of 2, there are 0 such Boolean algebras.
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60Let be a finite set. A relation on is defined as a quasi-order (or preorder) if it is reflexive and transitive. How can the quasi-order be naturally transformed into a proper partial order?
introduction
Hard
A.By computing the antisymmetric closure of .
B.It is mathematically impossible unless is inherently antisymmetric from the beginning.
C.By taking the quotient set of under the equivalence relation where .
D.By removing all symmetric pairs strictly from .
Correct Answer: By taking the quotient set of under the equivalence relation where .
Explanation:
A preorder lacks antisymmetry. Elements that violate antisymmetry (where and but ) form equivalence classes. By collapsing these equivalence classes via the quotient set, the resulting relation over the equivalence classes becomes strictly antisymmetric, thereby forming a valid partial order.