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web.groovymark@gmail.com
- November 9, 2024
Question 01
What is the symbol for the set of rational numbers?
a) R
b) N
c) Z
d) Q
Correct Answer: d) Q
Explanation: The symbol Q denotes the set of rational numbers, which includes all numbers that can be expressed as the quotient of two integers, with the denominator not equal to zero.
Question 02
Which statement describes the empty set?
a) It contains one element
b) It is a subset of every set
c) It cannot be a subset of any set
d) It contains multiple elements
Correct Answer: b) It is a subset of every set
Explanation: The empty set, denoted as ∅, is considered a subset of every set by definition, as it contains no elements to violate the subset condition.
Question 03
What does the notation A⊆B indicate?
a) A is a proper subset of B
b) A is not a subset of B
c) A is a subset of B
d) A is equal to B
Correct Answer: c) A is a subset of B
Explanation: The notation A⊆B signifies that every element of set A is also an element of set B, indicating a subset relationship.
Question 04
In set theory, what does the term “cardinality” refer to?
a) The maximum number of elements in a set
b) The minimum number of elements in a set
c) The number of elements in a set
d) The average number of elements across all sets
Correct Answer: c) The number of elements in a set
Explanation: Cardinality quantifies the size of a set by counting the number of elements it contains.
Question 05
Which law states that the intersection of a set with the empty set is always the empty set?
a) Domination Law
b) Absorption Law
c) Identity Law
d) Complement Law
Correct Answer: a) Domination Law
Explanation: The Domination Law asserts that intersecting any set with the empty set results in the empty set, as there are no elements to share.
Question 06
What is the result of the operation A∪(A∩B)?
a) B
b) A
c) A \cap B
d) A \cup B
Correct Answer: b) A
Explanation: The Absorption Law states that the union of a set A with the intersection of A and B will always return A, as all elements of A are retained.
Question 07
How is the symmetric difference between two sets A and B defined?
a) Elements common to both A and B
b) Elements that are in either A or B but not in both
c) Elements that are in A but not in B
d) The union of A and B
Correct Answer: b) Elements that are in either A or B but not in both
Explanation: The symmetric difference consists of elements that are in one set or the other but not in their intersection, effectively excluding shared elements.
Question 08
Which operation in set theory combines all elements from both sets while removing duplicates?
a) Intersection
b) Union
c) Symmetric difference
d) Power set
Correct Answer: b) Union
Explanation: The union operation combines all unique elements from both sets, ensuring that no element appears more than once in the resulting set.
Question 09
In graph theory, what does a “walk” refer to?
a) A sequence of edges with no restrictions
b) A closed path
c) A sequence of edges without repeating any vertex
d) A path that starts and ends at the same vertex
Correct Answer: a) A sequence of edges with no restrictions
Explanation: A walk in graph theory can traverse edges and vertices without any limitations on repetition, making it the most general type of traversal.
Question 10
What is the implication of a relation being symmetric?
a) If xRy, then yRx must also hold
b) If xRy and yRz, then xRz must also hold
c) Every element relates to itself
d) No pairs relate
Correct Answer: a) If xRy, then yRx must also hold
Explanation: A symmetric relation requires that for any relationship established between two elements, the reverse relationship must also be true.
Question 11
What does the symbol AC typically represent?
a) The power set of A
b) The complement of A
c) The intersection of A with the universal set
d) The union of A with the universal set
Correct Answer: b) The complement of A
Explanation: The notation AC is used to indicate the complement of set A, which includes all elements not found in A but present in the universal set.
Question 12
What is a key characteristic of a transitive relation?
a) It is symmetric
b) If xRy and yRz, then xRz must also hold
c) Every element is related to itself
d) No element relates to itself
Correct Answer: b) If xRy and yRz, then xRz must also hold
Explanation: A transitive relation maintains that if one element is related to a second, which in turn is related to a third, then the first element must also relate to the third.
Question 13
What does the term “indegree” refer to in the context of directed graphs?
a) The number of edges pointing into a vertex
b) The number of edges pointing out of a vertex
c) The total number of edges in the graph
d) The number of vertices connected
Correct Answer: a) The number of edges pointing into a vertex
Explanation: The indegree of a vertex counts how many directed edges are arriving at that vertex, providing insight into its connectivity.
Question 14
What does a function being “injective” mean?
a) It can map multiple elements to the same element
b) It maps each element in the domain to a unique element in the codomain
c) It has no inputs
d) It can only map to the same output
Correct Answer: b) It maps each element in the domain to a unique element in the codomain
Explanation: An injective function ensures that no two different elements in the domain map to the same element in the codomain, maintaining uniqueness.
Question 15
What is the Cartesian product of two sets A and B?
a) The set of all elements in A
b) The union of A and B
c) The set of all ordered pairs (a,b) where a∈A and b∈B
d) The intersection of A and B
Correct Answer: c) The set of all ordered pairs (a,b) where a∈A and b∈B
Explanation: The Cartesian product A×B is formed by pairing each element from set A with each element from set B, resulting in a set of ordered pairs.
Question 16
What does the notation f:A→B indicate?
a) A function from set B to set A
b) A function from set A to set B
c) A relation between two sets
d) An equivalence class
Correct Answer: b) A function from set A to set B
Explanation: The notation f:A→B denotes that fff is a function mapping elements from set A to elements in set B.
Question 17
What does the expression x∈A mean?
a) x is not an element of A
b) x is an element of A
c) x is a subset of A
d) x is equal to A
Correct Answer: b) x is an element of A
Explanation: The expression x∈Ax \in Ax∈A indicates that xxx belongs to set A, establishing membership within the set.
Question 18
In logic, what does the contrapositive of the statement “if p then q” represent?
a) If q, then p
b) If not p, then not q
c) If not q, then not p
d) If p, then not q
Correct Answer: c) If not q, then not p
Explanation: The contrapositive of a conditional statement "if p then q" is "if not q then not p," and it is logically equivalent to the original statement.
Question 19
What is the significance of a tautology in logic?
a) It can be false
b) It can have varying truth values
c) It is always true regardless of circumstances
d) It depends on the context
Correct Answer: c) It is always true regardless of circumstances
Explanation: A tautology is a logical statement that remains true in every possible scenario, regardless of the truth values of its components.
Question 20
What does the negation of a conjunction state according to De Morgan’s laws?
a) ¬(p∧q) = ¬p∧¬q
b) ¬(p∧q) = ¬p∨¬q
c) ¬(p∨q) = ¬p∧¬q
d) ¬(p∨q) = ¬p∨¬q
Correct Answer: b) ¬(p∧q) = ¬p∨¬q
Explanation: De Morgan's laws specify that the negation of a conjunction is equivalent to the disjunction of the negations, providing a key transformation in logical expressions.
