Secondary 4 E-Maths sample questions with answers

Set Language and Notation

1. 1ξ = {x : x is an integer, 1 ≤ x ≤ 15} and A = {x : x is a multiple of 4}. (a) List the elements of A. (b) Find n(A').

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   Answer: (a) {4, 8, 12} (b) 12

   1. (a) A = {4, 8, 12}
   2. (b) n(A') = 15 - 3 = 12

2. 2M = {x : x is a prime number, 30 < x < 50}. (a) List the elements of M. (b) Write down n(M).

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   Answer: (a) {31, 37, 41, 43, 47} (b) 5

   1. (a) M = {31, 37, 41, 43, 47}
   2. (b) n(M) = 5

3. 3P = {2, 5, 8, 11}. State whether each of these statements is true or false. (a) {5} ∈ P (b) {2, 11} ⊂ P

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   Answer: (a) False (b) True

   1. (a) False: {5} is a set, not an element. The correct statements are 5 ∈ P or {5} ⊂ P.
   2. (b) True: 2 and 11 are in P, and P has other elements (5 and 8).

4. 4n(ξ) = 24 and n(A') = 9. Find n(A).

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   Answer: 15

   1. n(A) + n(A') = n(ξ)
   2. n(A) = 24 - 9 = 15

5. 5ξ = {a, b, c, d, e, f, g}, X = {a, c, e, g} and Y = {b, c, d}. List the elements of (a) X ∩ Y, (b) X' ∩ Y'.

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   Answer: (a) {c} (b) {f}

   1. (a) X ∩ Y = {c}
   2. (b) X' = {b, d, f} and Y' = {a, e, f, g}, so X' ∩ Y' = {f}

6. 6P = {x : x is a prime number, 10 < x < 30}. (a) List the elements of P. (b) Find n(P). (c) Is 21 an element of P? Write your answer using ∈ or ∉.

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   Answer: (a) P = {11, 13, 17, 19, 23, 29} (b) n(P) = 6 (c) 21 ∉ P

   1. (a) The prime numbers between 10 and 30 are 11, 13, 17, 19, 23 and 29. So P = {11, 13, 17, 19, 23, 29}.
   2. (b) P has 6 elements, so n(P) = 6.
   3. (c) 21 = 3 × 7, so 21 is not prime. Hence 21 ∉ P.

7. 7Q = {letters of the word GEOGRAPHY}. List the elements of Q and find n(Q).

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   Answer: Q = {G, E, O, R, A, P, H, Y}, n(Q) = 8

   1. Each letter is written once, so G appears once.
   2. Q = {G, E, O, R, A, P, H, Y} and n(Q) = 8

8. 8ξ = {x : x is an integer, 1 ≤ x ≤ 12}, A = {x : x is a multiple of 3} and B = {x : x is a multiple of 13}. (a) List the elements of A'. (b) Find n(A'). (c) Describe the set B.

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   Answer: (a) A' = {1, 2, 4, 5, 7, 8, 10, 11} (b) 8 (c) B is the empty set

   1. (a) A = {3, 6, 9, 12}. A' is every other element of ξ: A' = {1, 2, 4, 5, 7, 8, 10, 11}.
   2. (b) n(A') = 8. Check: n(A) + n(A') = 4 + 8 = 12 = n(ξ).
   3. (c) No integer from 1 to 12 is a multiple of 13, so B has no elements. B is the empty set.

9. 9ξ = {x : x is an integer, 20 ≤ x ≤ 30} and S = {x : x is a perfect square}. Find n(S').

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   Answer: 10

   1. n(ξ) = 11 and S = {25}
   2. n(S') = 11 - 1 = 10

10. 10S = {2, 4, 6, 8}, T = {x : x is an even number, 1 ≤ x ≤ 9} and U = {4, 8, 12}. State whether each statement is true or false. (a) S ⊆ T (b) S ⊂ T (c) U ⊈ S (d) {4, 8} ⊂ U

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    Answer: (a) True (b) False (c) True (d) True

    1. List T first: T = {2, 4, 6, 8}, so S and T have exactly the same elements and S = T.
    2. (a) Every element of S is in T, so S ⊆ T is true.
    3. (b) T has no element that is not in S, so S is not a proper subset of T. S ⊂ T is false.
    4. (c) 12 ∈ U but 12 ∉ S, so U is not a subset of S. U ⊈ S is true.
    5. (d) 4 and 8 are both in U, and U also has 12. So {4, 8} ⊂ U is true.

11. 11List all the subsets of {p, q}. Which of them are proper subsets of {p, q}?

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    Answer: The subsets are ∅, {p}, {q} and {p, q}. The proper subsets are ∅, {p} and {q}.

    1. The empty set is a subset of every set, and every set is a subset of itself.
    2. No set is a proper subset of itself, so {p, q} is left out of the proper subsets.