Further Mathematics — Algebra — Sets, Surds, Indices & Polynomials
This is the real Level 1 question set — 20 questions, free, no account needed.
Tap a question to check your answer and read the explanation.
1. For any set A, what is A ∩ ∅ (the intersection of A with the empty set)?
A. A
B. ∅
C. U (the universal set)
D. A'
Explanation: The empty set has no elements, so it cannot share any elements with A. Therefore A ∩ ∅ = ∅ for every set A.
2. If a set A has 5 elements, how many elements are in the power set of A, P(A)?
A. 10
B. 16
C. 25
D. 32
Explanation: The power set of a set with n elements has 2^n elements. For n = 5, this is 2^5 = 32.
3. In a class of 50 students at a secondary school in Lagos, 30 students like Mathematics, 25 like Physics, and 10 like both subjects. How many students like neither Mathematics nor Physics?
A. 5
B. 10
C. 15
D. 20
Explanation: n(M ∪ P) = n(M) + n(P) - n(M ∩ P) = 30 + 25 - 10 = 45. Students who like neither = total - n(M ∪ P) = 50 - 45 = 5.
4. Given A = {1, 2, 3} and B = {2, 3, 4}, find the symmetric difference A Δ B (elements in A or B, but not in both).
A. {2, 3}
B. {1, 4}
C. {1, 2, 3, 4}
D. {1, 2, 3}
Explanation: The symmetric difference A Δ B = (A - B) ∪ (B - A). Here A - B = {1} and B - A = {4}, so A Δ B = {1, 4}.
5. Let A be the set of all factors of 12. Find n(A), the number of elements in A.
A. 4
B. 5
C. 6
D. 7
Explanation: The factors of 12 are 1, 2, 3, 4, 6, and 12, giving a total of 6 elements.
6. Which of the following correctly represents the set of even numbers less than 10 using set-builder notation?
A. {x : x is even, x < 10}
B. {2, 4, 6, 8}
C. {x : x is odd, x < 10}
D. {x : x > 10}
Explanation: Set-builder notation describes a set by stating a rule that its members satisfy, using the form {x : condition on x}. Option B is roster notation, not set-builder notation, while C and D describe the wrong condition.
7. On the set of real numbers, is the operation of subtraction commutative?
A. Yes, because a - b = b - a for all real numbers a and b
B. No, because a - b is generally not equal to b - a
C. Yes, but only when a and b are both positive
D. Cannot be determined without specific values
Explanation: For an operation to be commutative, a * b must equal b * a for all values. Since 5 - 3 = 2 but 3 - 5 = -2, subtraction is not commutative.
8. A binary operation * is defined on the set of real numbers by a * b = ab + a + b. Find the identity element for this operation.
A. 1
B. -1
C. 0
D. 2
Explanation: We need a * e = a, so ae + a + e = a, which simplifies to ae + e = 0, i.e., e(a+1) = 0. For this to hold for all a, e must equal 0. Checking: a * 0 = 0 + a + 0 = a, confirming e = 0.
9. Given A = {1, 2} and B = {1, 2, 3, 4}, which of the following statements is true?
A. A ⊂ B
B. B ⊂ A
C. A = B
D. A ∩ B = ∅
Explanation: Every element of A (1 and 2) is also found in B, so A is a proper subset of B, written A ⊂ B. B is not a subset of A since B has extra elements (3 and 4) not in A.
10. Given the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 2, 3, 4}, find the complement of A, A'.
A. {5, 6, 7, 8, 9, 10}
B. {1, 2, 3, 4}
C. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
D. {}
Explanation: The complement of A, written A', consists of all elements in U that are not in A. Removing {1,2,3,4} from U leaves {5,6,7,8,9,10}.
11. A binary operation * is defined on the set of integers by a * b = a + b - 1. Is this operation associative?
A. No, associativity fails for some integers
B. Yes, (a * b) * c = a * (b * c) for all integers
C. It is associative only when a = b = c
D. Associativity cannot be tested for this operation
Explanation: (a*b)*c = (a+b-1)+c-1 = a+b+c-2, and a*(b*c) = a+(b+c-1)-1 = a+b+c-2. Since both expressions are equal, the operation is associative.
12. A binary operation * is defined on the set of integers by a * b = a + b - 1. Find the identity element, e, such that a * e = a for every integer a.
A. 0
B. -1
C. 1
D. 2
Explanation: We require a * e = a, so a + e - 1 = a, giving e - 1 = 0, hence e = 1.
13. Two sets A and B are said to be disjoint if:
A. A ∪ B = U
B. A ∩ B = ∅
C. A ⊂ B
D. n(A) = n(B)
Explanation: Disjoint sets have no elements in common, so their intersection is the empty set, A ∩ B = ∅.
14. If A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7}, find A ∪ B.
A. {4, 5}
B. {1, 2, 3, 4, 5, 6, 7}
C. {1, 2, 3}
D. {6, 7}
Explanation: The union A ∪ B contains every element that is in A, in B, or in both, without repeating common elements. This gives {1,2,3,4,5,6,7}.
15. If n(A) = 15, n(B) = 20 and n(A ∩ B) = 5, find n(A ∪ B).
A. 25
B. 30
C. 35
D. 40
Explanation: Using the formula n(A ∪ B) = n(A) + n(B) - n(A ∩ B), we get 15 + 20 - 5 = 30.
16. A set contains 4 distinct elements. How many subsets does the set have in total?
A. 8
B. 12
C. 16
D. 20
Explanation: A set with n elements has 2^n subsets (including the empty set and the set itself). For n = 4, the number of subsets is 2^4 = 16.
17. Given A = {2, 4, 6, 8} and B = {4, 8, 10}, find A - B (the set of elements in A but not in B).
A. {2, 6}
B. {4, 8}
C. {10}
D. {2, 4, 6, 8, 10}
Explanation: A - B contains elements found in A that do not appear in B. Removing 4 and 8 (which are in B) from A leaves {2, 6}.
18. According to De Morgan's law, (A ∪ B)' is equal to:
A. A' ∪ B'
B. A' ∩ B'
C. A ∩ B
D. A ∪ B
Explanation: De Morgan's law states that the complement of a union equals the intersection of the complements: (A ∪ B)' = A' ∩ B'.
19. A binary operation * is defined on the set of real numbers by a * b = a^2 + b^2. Is this operation commutative?
A. No, because a^2 + b^2 does not equal b^2 + a^2
B. Yes, because a^2 + b^2 = b^2 + a^2 for all real numbers
C. It depends on whether a and b are positive
D. Commutativity cannot be tested for squares
Explanation: Since addition of real numbers is commutative, a^2 + b^2 always equals b^2 + a^2, so the operation is commutative.
20. What is the roster (listing) form of the set {x : x is a natural number, x < 5}?
A. {0, 1, 2, 3, 4}
B. {1, 2, 3, 4}
C. {1, 2, 3, 4, 5}
D. {2, 3, 4}
Explanation: Natural numbers start from 1 (in the convention commonly used in Nigerian secondary schools). Those less than 5 are 1, 2, 3, and 4.
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