ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Sets, Functions and Groups (Set 1)

Take the ECAT Mathematics — Sets, Functions and Groups (Set 1): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review.…

What this test covers

30 ECAT Mathematics

Sample questions with answers

The first 5 questions of this test, with the correct answer and explanation.

Q1. How many subsets does the set {a, b, c} have?
- A. 6
- B. 9
- C. 8 ✓
- D. 3
Answer: C. 8
Explanation: A set with n elements has 2ⁿ subsets; 2³ = 8.
Q2. A set has 5 elements. The number of its proper subsets is:
- A. 31 ✓
- B. 10
- C. 32
- D. 25
Answer: A. 31
Explanation: Total subsets = 2⁵ = 32; excluding the set itself leaves 31 proper subsets.
Q3. If A = {1, 2, 3, 4} and B = {3, 4, 5}, then A − B is:
- A. {3, 4}
- B. {1, 2, 5}
- C. {5}
- D. {1, 2} ✓
Answer: D. {1, 2}
Explanation: A − B contains the elements of A that are not in B: {1, 2}.
Q4. By De Morgan's law, (A ∪ B)′ is equal to:
- A. A′ ∪ B′
- B. A′ ∩ B′ ✓
- C. (A ∩ B)′
- D. A ∩ B
Answer: B. A′ ∩ B′
Explanation: De Morgan's law: the complement of a union is the intersection of the complements.
Q5. If n(A) = 20, n(B) = 15 and n(A ∩ B) = 7, then n(A ∪ B) is:
- A. 28 ✓
- B. 35
- C. 22
- D. 42
Answer: A. 28
Explanation: n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 20 + 15 − 7 = 28.

The other 25 questions

Take the timed test (free, no sign-up) to see your score and the answer and explanation for every question.

Q6. In a class of 40 students, 25 like Mathematics, 20 like Physics and 8 like neither. How many like both subjects?
- A. 12
- B. 5
- C. 17
- D. 13
Q7. The power set of the empty set, P(∅), contains:
- A. no element
- B. exactly one element
- C. infinitely many elements
- D. two elements
Q8. If A = {1, 2, 3} and B = {2, 3, 4}, the symmetric difference A Δ B is:
- A. {2, 3}
- B. {1, 4}
- C. {1, 2, 3, 4}
- D. {1}
Q9. The conditional p → q is false only when:
- A. p is true and q is false
- B. both p and q are true
- C. both p and q are false
- D. p is false and q is true
Q10. The contrapositive of the conditional p → q is:
- A. ~p → q
- B. q → p
- C. ~q → ~p
- D. ~p → ~q
Q11. Which of the following statements is a tautology?
- A. p ∨ ~p
- B. p → ~p
- C. p ∧ ~p
- D. p ↔ ~p
Q12. If f(x) = 2x + 3, then f⁻¹(x) is:
- A. 2x − 3
- B. 1/(2x + 3)
- C. (x + 3)/2
- D. (x − 3)/2
Q13. If f(x) = x² and g(x) = x + 1, then (f∘g)(2) is:
- A. 5
- B. 4
- C. 6
- D. 9
Q14. Which of the following relations from {1, 2, 3} to {1, 2, 3} is a function?
- A. {(1, 1), (2, 2), (2, 3)}
- B. {(1, 2), (1, 3), (2, 1)}
- C. {(1, 2), (2, 3), (3, 2)}
- D. {(1, 3), (3, 1), (3, 3)}
Q15. If A has 2 elements and B has 3 elements, the number of different relations from A to B is:
- A. 6
- B. 32
- C. 8
- D. 64
Q16. The function f: ℝ → ℝ defined by f(x) = x² is:
- A. one-one and onto
- B. neither one-one nor onto
- C. one-one but not onto
- D. onto but not one-one
Q17. Which function from ℝ to ℝ is a bijection?
- A. f(x) = x³
- B. f(x) = |x|
- C. f(x) = x²
- D. f(x) = x² + 1
Q18. The domain of the real function f(x) = 1/(x² − 9) is:
- A. ℝ − {9}
- B. ℝ − {−3, 3}
- C. ℝ − {3}
- D. (−3, 3)
Q19. The range of f(x) = x² + 1, x ∈ ℝ, is:
- A. [0, ∞)
- B. (1, ∞)
- C. [1, ∞)
- D. ℝ
Q20. If a binary operation * on ℤ is defined by a * b = a + b + 1, the inverse of 3 under * is:
- A. −3
- B. −4
- C. −2
- D. −5
Q21. Which of the following is a group under ordinary addition?
- A. The set of natural numbers ℕ
- B. The set of positive rationals
- C. The set of integers ℤ
- D. The set of odd integers
Q22. In the group {1, −1, i, −i} under multiplication, the inverse of i is:
- A. −i
- B. i
- C. 1
- D. −1
Q23. In the group {1, ω, ω²} of cube roots of unity under multiplication, the inverse of ω is:
- A. 1
- B. −ω
- C. ω²
- D. ω
Q24. The set ℕ = {1, 2, 3, …} under addition is:
- A. a group
- B. a monoid but not a group
- C. not even a semigroup
- D. a semigroup but not a monoid
Q25. In any group G, for elements a and b, (ab)⁻¹ is equal to:
- A. b⁻¹a⁻¹
- B. a⁻¹b⁻¹
- C. ab
- D. ba
Q26. A group in which ab = ba for all elements a and b is called:
- A. a trivial group
- B. a semigroup
- C. a cyclic monoid
- D. an abelian group
Q27. For the set of integers ℤ under the operation of subtraction, which statement is correct?
- A. Neither closure nor associativity holds
- B. Closure holds but associativity fails
- C. Both closure and associativity hold
- D. Associativity holds but closure fails
Q28. If n(A) = 3 and n(B) = 4, the number of elements in A × B is:
- A. 81
- B. 64
- C. 12
- D. 7
Q29. If A ⊆ B, then A ∪ B is equal to:
- A. B
- B. A
- C. A ∩ B
- D. ∅
Q30. In roster form, the set {x | x ∈ ℕ and x² < 20} is:
- A. {0, 1, 2, 3, 4}
- B. {1, 2, 3}
- C. {1, 2, 3, 4, 5}
- D. {1, 2, 3, 4}