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Unit 3 - Practice Quiz

MTH401

1 Which formula correctly represents the Principle of Inclusion-Exclusion for two finite sets and ?

A.
B.
C.
D.

2 In a class of 50 students, 30 study Mathematics, 25 study Physics, and 15 study both. How many students study at least one of the subjects?

A. 35
B. 40
C. 55
D. 45

3 The Pigeonhole Principle states that if items are put into containers, with , then at least one container must contain:

A. Exactly 1 item
B. More than 1 item
C. No items
D. Exactly 2 items

4 According to the Generalized Pigeonhole Principle, if objects are placed into boxes, then there is at least one box containing at least how many objects?

A.
B.
C.
D.

5 How many people must be in a room to guarantee that at least two of them were born in the same month?

A. 12
B. 13
C. 24
D. 365

6 What is the minimum number of students required in a class to guarantee that at least 5 students have the same grade, assuming there are 4 possible grades (A, B, C, D)?

A. 16
B. 17
C. 20
D. 21

7 Let . Which of the following is a relation on ?

A.
B.
C.
D.

8 The Cartesian product of sets and , denoted , is defined as:

A.
B.
C.
D.

9 A relation on set is reflexive if:

A.
B.
C.
D.

10 A relation is symmetric if:

A.
B.
C.
D. for all

11 A relation is antisymmetric if:

A.
B.
C.
D. It is not symmetric

12 A relation is transitive if:

A.
B.
C.
D.

13 If set has elements, how many possible relations are there on ?

A.
B.
C.
D.

14 Let be a relation represented by the matrix . If is reflexive, what must be true about ?

A. All entries are 1
B. All diagonal entries are 1
C. The matrix is symmetric
D. All off-diagonal entries are 0

15 If the matrix represents a relation , then is symmetric if and only if:

A. is an identity matrix
B. (Transpose)
C. has all 1s on the diagonal
D. is a triangular matrix

16 Given relations and on a set , the composition is defined as:

A.
B.
C.
D.

17 If and are the boolean matrices representing relations and , what matrix represents the composition ?

A. (Boolean OR)
B. (Boolean AND)
C. (Boolean Product)
D. (Boolean Product)

18 The inverse of a relation , denoted , is defined as:

A.
B.
C.
D. The empty set

19 A relation on a set is an Equivalence Relation if it is:

A. Reflexive, Symmetric, and Transitive
B. Reflexive, Antisymmetric, and Transitive
C. Irreflexive, Symmetric, and Transitive
D. Symmetric and Transitive only

20 Which of the following is an equivalence relation on the set of integers ?

A.
B. is odd
C.
D. divides

21 The set of all elements related to an element in an equivalence relation is called:

A. The partition of
B. The equivalence class of , denoted
C. The power set of
D. The range of

22 Equivalence classes of a relation on set form a:

A. Partition of
B. Subset of
C. Product of
D. Lattice

23 A relation on a set is a Partial Ordering Relation (POSET) if it is:

A. Reflexive, Symmetric, Transitive
B. Reflexive, Antisymmetric, Transitive
C. Irreflexive, Antisymmetric, Transitive
D. Symmetric, Antisymmetric, Transitive

24 Which of the following is a classic example of a Partial Ordering Relation?

A. "is the father of" on a set of people
B. "is a subset of" () on a power set
C. "is friend of" on a set of people
D. "rock, paper, scissors" dominance

25 In a POSET , two elements and are called comparable if:

A.
B. or
C. and
D. They share an upper bound

26 A POSET in which every pair of elements is comparable is called a:

A. Lattice
B. Total Ordering (or Chain)
C. Equivalence Relation
D. Well-ordered set

27 What kind of graph is used to visualize a Partial Ordering Relation?

A. Scatter plot
B. Hasse Diagram
C. Pie Chart
D. Venn Diagram

28 In a Hasse diagram, an edge is drawn from to (where is below ) if:

A. and there is no such that
B.
C.
D. and are incomparable

29 In a POSET, an element is maximal if:

A. For all ,
B. There is no element such that
C. It is the only element in the set
D. It is greater than all other elements

30 An element in a POSET is the Greatest Lower Bound (GLB) of a subset if:

A. It is an upper bound and smaller than any other upper bound
B. It is a lower bound and greater than any other lower bound
C. It is the smallest element in the subset
D. It is the largest element in the subset

31 A POSET is a Lattice if:

A. Every pair of elements has a Least Upper Bound and a Greatest Lower Bound
B. It is a total ordering
C. It has a unique maximum element
D. Every subset has a least element

32 In the context of Lattices, the operation (Join) represents:

A. Greatest Lower Bound (GLB)
B. Least Upper Bound (LUB)
C. Set Intersection
D. Subtraction

33 In the context of Lattices, the operation (Meet) represents:

A. Greatest Lower Bound (GLB)
B. Least Upper Bound (LUB)
C. Set Union
D. Addition

34 A Lattice is called Bounded if:

A. It has a finite number of elements
B. It has both a greatest element ($1$) and a least element ($0$)
C. It satisfies the distributive law
D. It is a subset of real numbers

35 A Sublattice is a subset of a lattice that:

A. Is a lattice itself under any operation
B. Is closed under the Join () and Meet () operations of the original lattice
C. Contains the top and bottom elements
D. Is a chain

36 Which set with the relation 'divides' () forms a lattice?

A.
B. (Divisors of 6)
C.
D.

37 A relation is Irreflexive if:

A. for all
B. for some
C.
D. It is not reflexive

38 For the Principle of Inclusion-Exclusion with 3 sets, . What is ?

A.
B.
C.
D.

39 A Distributive Lattice is a lattice where:

A. holds
B. Every element has a complement
C. The order is total
D. implies

40 Which diagram represents the relation on the power set of ?

A. A triangle
B. A diamond (rhombus) shape
C. A straight line
D. A square with diagonals

41 The complement of a relation on set , denoted , is:

A.
B.
C.
D. The identity relation

42 Consider the relation on . Is it transitive?

A. Yes
B. No, because and are in , but is not
C. No, because it is symmetric
D. No, because is in

43 If is an equivalence relation on set , then the Quotient Set is:

A. The set of all equivalence classes
B. The set minus
C. The largest subset of
D. The intersection of all classes

44 In a Hasse diagram, if there is a path from to , then:

A. covers
B.
C.
D. and are equal

45 The relation 'less than' () on integers is:

A. Reflexive and Transitive
B. Irreflexive and Transitive
C. Symmetric and Transitive
D. Equivalence Relation

46 Let be a lattice. For any , ?

A. $0$
B. $1$
C.
D.

47 Consider a set . How many ordered pairs are in the Universal Relation on ?

A. 3
B. 6
C. 9
D.

48 Which of the following properties does the Identity Relation satisfy?

A. Only Reflexive
B. Only Symmetric
C. Equivalence Relation and Partial Order
D. None of the above

49 If we represent a relation using a directed graph (digraph), a reflexive relation is characterized by:

A. No cycles
B. A self-loop on every vertex
C. Bidirectional edges
D. No edges

50 The Principle of Inclusion-Exclusion is used to calculate the cardinality of the:

A. Union of sets
B. Intersection of sets
C. Power set
D. Cartesian product