ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Sets, Functions and Groups (Set 5): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review.…
What this test covers
The first 5 questions of this test, with the correct answer and explanation.
Q1. How many relations can be defined from a set with 2 elements to a set with 3 elements?
Answer: D. 64
Explanation: A relation from A to B is any subset of A × B, which has 6 elements, so there are 2⁶ = 64.
Q2. The number of onto functions from a 4-element set to a 3-element set is:
Answer: A. 36
Explanation: Total functions 3⁴ = 81. Subtract those missing at least one image: 3·2⁴ − 3·1 = 45. So 81 − 45 = 36 (equivalently C(4,2)·3! = 36).
Q3. The number of one-one functions from a 3-element set into a 5-element set is:
Answer: C. 60
Explanation: Images are chosen without repetition: 5 × 4 × 3 = 60.
Q4. The number of relations on {1, 2, 3} that are both reflexive and symmetric is:
Answer: A. 8
Explanation: The three pairs (a, a) must be present; the three unordered pairs {a, b} with a ≠ b can each be included or not, giving 2³ = 8.
Q5. The number of equivalence relations on a 3-element set is:
Answer: C. 5
Explanation: Equivalence relations correspond to partitions: one block (1 way), a 2 + 1 split (3 ways), three singletons (1 way), total 5.
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Q6. The domain of f(x) = √(x − 1)/(x² − 9) is:
Q7. The range of f(x) = x² − 4x + 7 over the real numbers is:
Q8. Let f(x) = 2x + 3 and g(x) = x². The value of (f∘g)(1) − (g∘f)(1) is:
Q9. If f(x) = (2x + 1)/(x − 3), then f⁻¹(x) is:
Q10. If f(x) = 1/(1 − x), then f(f(f(x))) equals:
Q11. By De Morgan's law, (A ∩ B′)′ equals:
Q12. If n(A) = 40, n(B) = 35 and n(A ∪ B) = 60, then n(A Δ B), the number of elements in exactly one of the sets, is:
Q13. In a group of 100 students, 45 study maths, 40 physics, 30 chemistry; 15 study maths and physics, 10 physics and chemistry, 12 maths and chemistry, and 5 study all three. How many study none of the three?
Q14. A set has 5 elements. The number of its subsets having at least 2 elements is:
Q15. How many subsets of {1, 2, 3, 4, 5}, including the empty set, contain no two consecutive integers?
Q16. In the truth table of (p ∨ q) → (p ∧ q), the number of rows with truth value T is:
Q17. The negation of the conditional statement p → q is logically equivalent to:
Q18. Which of the following compound statements is a contradiction (false for every truth assignment)?
Q19. The truth table of a compound statement containing 4 different simple statements has how many rows?
Q20. In the group (Z₁₂, +) of integers modulo 12, the order of the element 3 is:
Q21. In the group of non-zero residues modulo 13 under multiplication, the inverse of 5 is:
Q22. The number of generators of the cyclic group (Z₈, +) is:
Q23. On R \ {1}, define a ∗ b = a + b − ab. The inverse of 3 under ∗ is:
Q24. On the integers define a ∗ b = a + b + 1. The inverse of 5 is:
Q25. In the group {1, −1, i, −i} under multiplication, how many elements x satisfy x² = −1?
Q26. Which of these groups has the property that every element is its own inverse?
Q27. The number of elements in {x : x is an integer and x² < 20} is:
Q28. If A = {1, 2} and B = {2, 3}, then n((A × B) ∩ (B × A)) equals:
Q29. The function f: R → R defined by f(x) = x³ − x is:
Q30. The number of binary operations that can be defined on a set with 2 elements is: