Q.1 Which of the following is a tautology?
p ∧ ¬p
p ∨ ¬p
p → p
¬p ∧ p
Explanation - A tautology is a statement that is always true. p ∨ ¬p is always true regardless of the truth value of p.
Correct answer is: p ∨ ¬p
Q.2 What is the contrapositive of the statement 'If it rains, then the ground gets wet'?
If the ground gets wet, then it rains
If the ground does not get wet, then it does not rain
If it does not rain, then the ground does not get wet
If it rains, then the ground does not get wet
Explanation - The contrapositive of 'If P then Q' is 'If not Q then not P', which is logically equivalent to the original statement.
Correct answer is: If the ground does not get wet, then it does not rain
Q.3 Which set is countably infinite?
The set of natural numbers
The set of real numbers between 0 and 1
The set of irrational numbers
The set of points in a plane
Explanation - A set is countably infinite if its elements can be put in one-to-one correspondence with natural numbers. Real numbers and points in a plane are uncountable.
Correct answer is: The set of natural numbers
Q.4 Which of the following logical equivalences is correct?
¬(p ∧ q) ≡ ¬p ∨ ¬q
¬(p ∧ q) ≡ ¬p ∧ ¬q
¬(p ∨ q) ≡ ¬p ∨ ¬q
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ r
Explanation - This is De Morgan's Law, which relates the negation of conjunctions and disjunctions.
Correct answer is: ¬(p ∧ q) ≡ ¬p ∨ ¬q
Q.5 Which of the following is an example of a recursive definition?
Fibonacci sequence
Prime numbers
Odd numbers
Multiples of 5
Explanation - A recursive definition defines an object in terms of itself. Fibonacci numbers are defined recursively: F(n) = F(n-1) + F(n-2).
Correct answer is: Fibonacci sequence
Q.6 What is the truth value of the statement ∀x ∈ ℝ, x^2 ≥ 0?
True
False
Depends on x
Cannot be determined
Explanation - The square of any real number is always non-negative, so the universal statement is true.
Correct answer is: True
Q.7 Which of the following is not a valid inference rule in propositional logic?
Modus Ponens
Modus Tollens
Disjunctive Syllogism
Affirming the Consequent
Explanation - Affirming the consequent is a logical fallacy, not a valid inference rule.
Correct answer is: Affirming the Consequent
Q.8 Which type of proof assumes the negation of the statement to be proved?
Direct proof
Proof by induction
Proof by contradiction
Constructive proof
Explanation - In proof by contradiction, we assume the negation of the statement and show it leads to a contradiction.
Correct answer is: Proof by contradiction
Q.9 If p → q is true and p is false, what can we say about q?
q must be true
q must be false
q can be true or false
q is undefined
Explanation - An implication is true if the premise is false, regardless of the truth value of the conclusion.
Correct answer is: q can be true or false
Q.10 Which of the following represents an injective function?
f(x) = x^2 over ℝ
f(x) = 2x + 1 over ℝ
f(x) = sin(x) over ℝ
f(x) = x^3 - x over ℝ
Explanation - A function is injective if each element of the domain maps to a unique element in the codomain. f(x) = 2x + 1 is one-to-one.
Correct answer is: f(x) = 2x + 1 over ℝ
Q.11 What is the negation of the statement 'All swans are white'?
No swans are white
Some swans are not white
All swans are not white
Some swans are white
Explanation - The negation of a universal statement 'All X are Y' is 'There exists X that is not Y'.
Correct answer is: Some swans are not white
Q.12 Which of the following statements about sets is true?
The power set of a set with n elements has n elements
The power set of a set with n elements has 2^n elements
The union of two sets has fewer elements than either set
The intersection of two sets has more elements than either set
Explanation - The power set contains all possible subsets, which is 2^n for a set of n elements.
Correct answer is: The power set of a set with n elements has 2^n elements
Q.13 Which of the following is an example of a well-formed formula in propositional logic?
p ∧ (q ∨ r)
∧ p q
p ∨ ∨ q r
p q ∧ r
Explanation - A well-formed formula is syntactically correct; it must respect the rules of logical operators and parentheses.
Correct answer is: p ∧ (q ∨ r)
Q.14 Which of the following is true for an equivalence relation?
It is reflexive, symmetric, and transitive
It is only reflexive
It is only symmetric
It is only transitive
Explanation - An equivalence relation satisfies all three properties: reflexive, symmetric, and transitive.
Correct answer is: It is reflexive, symmetric, and transitive
Q.15 Which of the following statements is true about prime numbers?
2 is the only even prime number
All even numbers are prime
1 is a prime number
0 is a prime number
Explanation - A prime number has exactly two distinct positive divisors. 2 is the only even number satisfying this.
Correct answer is: 2 is the only even prime number
Q.16 What is the base case in mathematical induction?
The first step to prove the statement
The assumption step
The conclusion step
The recursive step
Explanation - The base case establishes the truth of the statement for the initial value, usually n=0 or n=1.
Correct answer is: The first step to prove the statement
Q.17 Which of the following is a valid disjunction?
p ∨ q
p ∧ q
p → q
¬p
Explanation - A disjunction is a logical 'or' statement represented as p ∨ q.
Correct answer is: p ∨ q
Q.18 Which of the following statements is false?
Every finite set is countable
Every infinite set is uncountable
The set of integers is countable
The empty set is finite
Explanation - Some infinite sets, like natural numbers or integers, are countable. Not all infinite sets are uncountable.
Correct answer is: Every infinite set is uncountable
Q.19 Which of the following is an example of a bijective function?
f(x) = x^2 over ℝ
f(x) = x + 1 over ℝ
f(x) = sin(x) over ℝ
f(x) = x^3 over ℝ
Explanation - A bijective function is both injective and surjective. f(x) = x^3 is one-to-one and onto ℝ.
Correct answer is: f(x) = x^3 over ℝ
Q.20 Which of the following statements is logically equivalent to p → q?
¬p ∨ q
¬q ∨ p
p ∧ q
¬p ∧ ¬q
Explanation - The implication p → q is logically equivalent to ¬p ∨ q.
Correct answer is: ¬p ∨ q
Q.21 Which of the following is true about the union of sets A and B?
Contains elements in both A and B
Contains elements in A but not in B
Contains elements in B but not in A
Contains only elements common to A and B
Explanation - The union of sets contains all elements that are in A, or in B, or in both.
Correct answer is: Contains elements in both A and B
Q.22 Which of the following is a valid method to prove '∀n ∈ ℕ, P(n)'?
Direct proof
Proof by induction
Proof by contradiction
All of the above
Explanation - Universal statements can be proved by direct proof, induction, or contradiction.
Correct answer is: All of the above
Q.23 What is the main property of an antisymmetric relation?
If (a, b) and (b, a) are in the relation, then a = b
If (a, b) is in the relation, then (b, a) must be in it
Every element relates to itself
Every element relates to all others
Explanation - Antisymmetry states that the only way both (a, b) and (b, a) can exist in the relation is if a = b.
Correct answer is: If (a, b) and (b, a) are in the relation, then a = b
Q.24 Which of the following is a well-ordering principle?
Every non-empty set of natural numbers has a least element
Every set of integers has a greatest element
Every finite set of real numbers has no minimum
All sets have a minimum
Explanation - This is the well-ordering principle, fundamental in proofs by induction.
Correct answer is: Every non-empty set of natural numbers has a least element
Q.25 Which of the following statements about Cartesian products is correct?
A × B = B × A always
The Cartesian product A × B contains ordered pairs (a, b) where a ∈ A and b ∈ B
A × ∅ = A
The cardinality of A × B is |A| + |B|
Explanation - The Cartesian product forms all ordered pairs from A and B. Order matters, and ∅ yields an empty product.
Correct answer is: The Cartesian product A × B contains ordered pairs (a, b) where a ∈ A and b ∈ B