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Unit 1: Discrete Mathematics - Practice Quiz

CSE333 — Combinatorial Studies-I 60 Questions
0 Correct 0 Wrong 60 Left
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1 Which of the following is a proposition?

propositional logic Easy
A. What time is it?
B.
C. Close the door.
D. Please sit down.

2 What is the negation of the proposition "It is raining"?

propositional logic Easy
A. It is cloudy.
B. It is sunny.
C. It may be raining.
D. It is not raining.

3 The conjunction is true when:

propositional logic Easy
A. Both and are true.
B. At least one of and is true.
C. Exactly one of and is true.
D. Both and are false.

4 Which symbol is commonly used for the universal quantifier?

first order logic Easy
A.
B.
C.
D.

5 Which symbol is commonly used for the existential quantifier?

first order logic Easy
A.
B.
C.
D.

6 If , which statement is true?

sets Easy
A.
B.
C.
D.

7 What is the cardinality of the set ?

sets Easy
A.
B.
C.
D.

8 What is the union of and ?

sets Easy
A.
B.
C.
D.

9 A relation from a set to a set is a subset of:

relations Easy
A.
B.
C.
D.

10 A relation on a set is reflexive if:

relations Easy
A. $ for every $a, b \in A$
B. implies
C. for every
D. and imply

11 A relation is symmetric if:

relations Easy
A. implies
B. for every
C. implies
D. and imply

12 A function from to assigns each element of to:

functions Easy
A. No elements of
B. Every element of
C. Exactly one element of
D. At least two elements of

13 The set of input values of a function is called its:

functions Easy
A. Image
B. Range
C. Domain
D. Codomain

14 A function is one-to-one if:

functions Easy
A. All inputs have the same output.
B. The domain and codomain are equal.
C. Different inputs have different outputs.
D. Every output has several inputs.

15 A partial order must be reflexive, antisymmetric, and:

partial orders Easy
A. Transitive
B. Invertible
C. Symmetric
D. Periodic

16 Under the usual order on the integers, which pair is comparable?

partial orders Easy
A. and
B. and
C. and
D. and

17 In a lattice, every pair of elements has a unique:

lattices Easy
A. Maximum only
B. Inverse and identity
C. Least upper bound and greatest lower bound
D. Minimum only

18 The least upper bound of two elements is also called their:

lattices Easy
A. Join
B. Complement
C. Inverse
D. Meet

19 Which property requires that a binary operation on a group be closed?

groups Easy
A. Every group element has a different name.
B. The group must contain exactly two elements.
C. The operation must be commutative.
D. The result of two group elements is in the group.

20 Every group must contain an element called the:

groups Easy
A. Largest element
B. Repeated element
C. Zero element
D. Identity element

21 Which proposition is logically equivalent to ?

propositional logic Medium
A.
B.
C.
D.

22 The formula is false for which assignment?

propositional logic Medium
A.
B.
C.
D.

23 From the premises , , and , which conclusion follows?

propositional logic Medium
A.
B.
C.
D.

24 Let mean " is a student" and mean " passed." Which formula states that exactly one student passed?

first order logic Medium
A.
B.
C.
D.

25 What is the negation of ?

first order logic Medium
A.
B.
C.
D.

26 Given , , and , which statement must be true?

first order logic Medium
A.
B.
C.
D.

27 In a class of 60 students, 35 study algebra, 28 study combinatorics, and 15 study both. How many study neither subject?

sets Medium
A.
B.
C.
D.

28 Let and . How many subsets of contain every element of ?

sets Medium
A.
B.
C.
D.

29 Define a relation on the integers by if and only if is divisible by . Which description of is correct?

relations Medium
A. It is reflexive and transitive but not symmetric.
B. It is an equivalence relation with four classes.
C. It is reflexive and symmetric but not transitive.
D. It is symmetric and transitive but not reflexive.

30 Let and . Which pair belongs to , where when some satisfies and ?

relations Medium
A.
B.
C.
D.

31 A relation on is reflexive and contains and . If is also symmetric and transitive, which pair must it contain?

relations Medium
A. only
B. only
C.
D. only

32 For the function defined by , which property holds?

functions Medium
A. It is surjective but not injective.
B. It is injective but not surjective.
C. It is both injective and surjective.
D. It is neither injective nor surjective.

33 Let be defined by and by . What is ?

functions Medium
A.
B.
C.
D.

34 How many surjective functions are there from a three-element set to a two-element set?

functions Medium
A.
B.
C.
D.

35 Consider the set ordered by divisibility. Which statement is correct?

partial orders Medium
A. is the greatest element.
B. is the least element.
C. is the unique minimal element.
D. is the unique minimal element.

36 In the poset , what is the least upper bound of and ?

partial orders Medium
A.
B.
C.
D.

37 In the lattice of positive divisors of ordered by divisibility, what are the meet and join of and , respectively?

lattices Medium
A. and
B. and
C. and
D. and

38 In the lattice , which operations represent meet, join, and complement?

lattices Medium
A. Intersection, difference, and union
B. Union, intersection, and set difference
C. Difference, union, and intersection
D. Intersection, union, and relative complement

39 In the additive group , what is the order of the element ?

groups Medium
A.
B.
C.
D.

40 Let and in . Using right-to-left composition, what is ?

groups Medium
A.
B.
C.
D.

41 How many truth assignments satisfy the formula ?

propositional logic Hard
A.
B.
C.
D.

42 From the premises , , and , which conclusion is logically necessary?

propositional logic Hard
A.
B.
C.
D.

43 Which set of connectives is functionally complete, meaning that every Boolean function can be expressed using only connectives from that set?

propositional logic Hard
A.
B.
C.
D.

44 Let mean " is strictly greater than ." Which formula states that every element has a strictly greater element, while no element is strictly greater than itself?

first order logic Hard
A.
B.
C.
D.

45 Consider the sentence Over structures with a nonempty domain, which statement is correct?

first order logic Hard
A. It has both finite and infinite models.
B. It has infinite models but no finite models.
C. It has finite models but no infinite models.
D. It has no models of any cardinality.

46 Assuming nonempty domains, which formula is logically equivalent to ?

first order logic Hard
A.
B.
C.
D.

47 Let be countably infinite, and let be the set of all finite subsets of . What is the cardinality of ?

sets Hard
A.
B.
C.
D.

48 For finite sets , suppose , , , , , , and . How many elements belong to exactly one of the three sets?

sets Hard
A.
B.
C.
D.

49 Define a relation on by if and only if or . Which description of is correct?

relations Hard
A. It is symmetric and transitive but not reflexive.
B. It is an equivalence relation with six classes.
C. It is an equivalence relation with four classes.
D. It is reflexive and symmetric but not transitive.

50 On , let . How many ordered pairs are in the transitive closure , where paths must have positive length?

relations Hard
A.
B.
C.
D.

51 How many surjective functions exist from a -element set onto a -element set?

functions Hard
A.
B.
C.
D.

52 Let and satisfy . Which conclusion must hold?

functions Hard
A. is necessarily .
B. is surjective and is injective.
C. is injective and is surjective.
D. Both and are necessarily bijective.

53 Consider the positive divisors of ordered by divisibility. How many maximal chains run from to ?

partial orders Hard
A.
B.
C.
D.

54 A poset on has only the nontrivial constraints , , , and , with incomparable to every other element. How many linear extensions does it have?

partial orders Hard
A.
B.
C.
D.

55 What is the maximum possible size of an antichain in the Boolean poset ?

partial orders Hard
A.
B.
C.
D.

56 In the lattice of positive divisors of ordered by divisibility, with meet and join , how many elements have a complement?

lattices Hard
A.
B.
C.
D.

57 Which combination of properties correctly describes the five-element diamond lattice ?

lattices Hard
A. Modular, distributive, and uncomplemented
B. Nonmodular, complemented, and distributive
C. Modular, complemented, and nondistributive
D. Distributive, complemented, and nonmodular

58 How many group homomorphisms have an image of order exactly ?

groups Hard
A.
B.
C.
D.

59 How many permutations satisfy ?

groups Hard
A.
B.
C.
D.

60 Let be a nonabelian group of order . How many elements of order does contain?

groups Hard
A.
B.
C.
D.
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Unit 2