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web.groovymark@gmail.com
- November 9, 2024
Question 01
What does it mean for two sets A and B to be equal?
a) They contain the same elements
b) They have different cardinalities
c) They are disjoint sets
d) One is a subset of the other
Correct Answer: a) They contain the same elements
Explanation: Two sets A and B are equal if every element in A is also in B and vice versa, indicating that they are identical in terms of their elements.
Question 02
What is the definition of a rational number?
a) Any number that can be expressed as a fraction a\b where a and b are integers and b ≠ 0
b) Any integer
c) Any decimal number
d) Any whole number
Correct Answer: a) Any number that can be expressed as a fraction a=b where a and b are integers and b ≠ 0
Explanation: A rational number is defined as any number that can be expressed in the form of a fraction with integer numerator and denominator, where the denominator is not zero.
Question 03
In set theory, what is the definition of a universal set?
a) A set that contains no elements
b) A set that includes all possible elements under consideration
c) A set that is a subset of another set
d) A set that can only contain rational numbers
Correct Answer: b) A set that includes all possible elements under consideration
Explanation: The universal set encompasses all elements relevant to a particular discussion or problem, serving as the context for defining subsets.
Question 04
What does the symbol ∀ represent in logical expressions?
a) There exists
b) For all
c) There is at least one
d) None of the above
Correct Answer: b) For all
Explanation: The symbol ∀ denotes universal quantification, indicating that a statement applies to all elements within a specified domain.
Question 05
What does the term “disjoint” refer to in set theory?
a) Sets that share some elements
b) Sets that are equal
c) Sets that have no elements in common
d) Sets that are subsets of each other
Correct Answer: c) Sets that have no elements in common
Explanation: Disjoint sets are defined as sets that do not share any elements, meaning their intersection is empty.
Question 06
What does the notation P(A) refer to?
a) The product of set A
b) The power set of A
c) The set of elements in A
d) The intersection of A
Correct Answer: b) The power set of A
Explanation: The notation P(A) represents the power set of A, which includes all possible subsets of the set A.
Question 07
In logic, what is a contradiction?
a) A statement that is always true
b) A statement that is sometimes true
c) A statement that is always false
d) A statement that can be true or false
Correct Answer: c) A statement that is always false
Explanation: A contradiction is defined as a logical statement that cannot be true under any circumstances, consistently yielding a false outcome.
Question 08
What does the notation AC represent?
a) The intersection of A
b) The union of A
c) The complement of A
d) The power set of A
Correct Answer: c) 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 09
What is the primary function of a relation in mathematics?
a) To provide a numeric value
b) To establish a connection between elements of two sets
c) To define an operation
d) To limit the elements in a set
Correct Answer: b) To establish a connection between elements of two sets
Explanation: A relation serves to define how elements from one set correspond to elements in another set, establishing connections between them.
Question 10
What does the symbol ∃ represent in logical expressions?
a) For all
b) There exists
c) Not
d) And
Correct Answer: b) There exists
Explanation: The symbol ∃ denotes existential quantification, indicating that there is at least one element in the domain for which a given predicate is true.
Question 11
Which property describes a reflexive relation?
a) No element relates to itself
b) Every element relates to itself
c) Some elements are unrelated
d) All elements relate to at least one other element
Correct Answer: b) Every element relates to itself
Explanation: A reflexive relation requires that every element in the set has a self-loop, meaning that for each element x, xRx holds true.
Question 12
What is a key feature of a surjective function?
a) Every element in the codomain must have a unique pre-image
b) At least one element in the codomain must not be covered
c) Every element in the codomain is mapped by at least one element in the domain
d) It cannot have an inverse
Correct Answer: c) Every element in the codomain is mapped by at least one element in the domain
Explanation: A surjective function ensures that all elements of the codomain are represented by at least one element from the domain, guaranteeing complete coverage.
Question 13
What does the expression A\B denote?
a) The union of A and B
b) The intersection of A and B
c) The elements in A that are not in B
d) The symmetric difference of A and B
Correct Answer: c) The elements in A that are not in B
Explanation: The notation A\B indicates the difference between sets A and B, representing the elements in A that do not belong to B.
Question 14
What does a proof by cases involve?
a) Considering all scenarios separately to validate a statement
b) Finding a contradiction
c) Assuming the conclusion is true
d) Proving one statement
Correct Answer: a) Considering all scenarios separately to validate a statement
Explanation: Proof by cases requires analyzing various possible scenarios or conditions that might affect the truth of the statement and proving each one individually.
Question 15
What does the term “cartesian product” refer to in set theory?
a) The union of two sets
b) The intersection of two sets
c) The set of all ordered pairs formed by two sets
d) The set of subsets of a set
Correct Answer: c) The set of all ordered pairs formed by two sets
Explanation: The Cartesian product of two sets A and B, denoted A×B, consists of all possible ordered pairs (a,b) where a∈A and b∈B.
Question 16
What does the term “idempotent law” in Boolean algebra imply?
a) Combining a variable with itself yields the variable
b) The AND operation will always yield true
c) The OR operation can be simplified to false
d) Variables cannot repeat
Correct Answer: a) Combining a variable with itself yields the variable
Explanation: The idempotent law states that both p∨p = p and p∧p = p, indicating redundancy in combining a variable with itself.
Question 17
What does the negation of a disjunction represent 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: a) ¬(p∨q) = ¬p∧¬q
Explanation: According to De Morgan's laws, the negation of a disjunction is equivalent to the conjunction of the negations, allowing for logical transformations.
Question 18
What does the term “equivalence relation” mean?
a) A relation that is reflexive, symmetric, and transitive
b) A relation that is only reflexive
c) A relation with no elements
d) A relation that is asymmetric
Correct Answer: a) A relation that is reflexive, symmetric, and transitive
Explanation: An equivalence relation must satisfy all three properties: reflexivity, symmetry, and transitivity, allowing for the classification of elements into equivalence classes.
Question 19
What is the main outcome of A∩BC?
a) The intersection of A and B
b) The elements in A that are not in B
c) The union of A and B
d) The empty set
Correct Answer: b) The elements in A that are not in B
Explanation: The operation A∩BC yields the elements in set A that are not present in set B, effectively isolating A's unique elements.
Question 20
What does the expression p∨¬p signify in logic?
a) It is a contradiction
b) It is always false
c) It is a tautology
d) It is dependent on the context
Correct Answer: c) It is a tautology
Explanation: The expression p∨¬p is a tautology because it will always yield true regardless of the truth value of p.
