Sec 3 E-Math Set Language & Notation topical questions

List the elements of A = {x : x is an even number and 1 ≤ x ≤ 11}, and state n(A).

State whether each statement is true or false: (a) 9 ∈ {multiples of 3}; (b) 10 ∈ {multiples of 3}.

Let ξ = {1, 2, 3, …, 12} and A = {x : x is a factor of 12}. (a) List the elements of A. (b) State n(A). (c) Is A ⊆ ξ? Give a reason.

1. (a)List A.[1]
2. (b)State n(A).[1]
3. (c)Is A ⊆ ξ?[1]

Given A = {2, 4, 6} and B = {4, 6, 8}, find (a) A ∩ B, (b) A ∪ B, and (c) n(A ∪ B).

On a two-circle Venn diagram, describe (in words) which regions are shaded for (a) A ∪ B, (b) A ∩ B, (c) A′.

The universal set is the integers from 1 to 15. E = {even numbers} and M = {multiples of 3}. List (a) E ∩ M, (b) E ∩ M′, and state (c) n(E ∪ M).

On a three-set Venn diagram (sets A, B and C), describe the region represented by (A ∩ B) ∪ C.

The universal set is ξ = {1, 2, …, 15}. P = {multiples of 3} and Q = {multiples of 5}. (a) List P ∩ Q. (b) List P ∪ Q. (c) Verify n(P ∪ Q) using the formula n(P) + n(Q) − n(P ∩ Q).

1. (a)List P ∩ Q.[1]
2. (b)List P ∪ Q.[1]
3. (c)Verify with the formula.[2]

In a year group of 40 students, 26 study Physics, 22 study Chemistry and 10 study neither subject. (a) Find the number studying at least one of the two subjects. (b) Find the number studying both.

1. (a)Number studying at least one.[2]
2. (b)Number studying both.[2]

In a class of 35 students, 20 play football (F), 15 play basketball (B) and 8 play both. (a) State the number in each region of the Venn diagram. (b) How many play exactly one sport? (c) How many play neither?

1. (a)All four regions.[2]
2. (b)Exactly one sport.[1]
3. (c)Neither sport.[1]

Given ξ = {1, 2, …, 10} and A = {2, 3, 5, 7} (the primes), find (a) A′ and (b) n(A′). (c) Check that n(A) + n(A′) = n(ξ).

Let M = {1, 2, 3}. Determine whether each statement is true or false: (a) {1} ⊂ M; (b) ∅ ⊆ M; (c) M ⊂ M.

1. (a){1} ⊂ M.[1]
2. (b)∅ ⊆ M.[1]
3. (c)M ⊂ M.[2]

Given n(A) = 17, n(B) = 14 and n(A ∩ B) = 5, with n(ξ) = 36, find (a) n(A ∪ B), (b) the number in A but not B, (c) the number in B but not A, and (d) the number outside both sets.

Of 40 travellers, 25 visited Japan (J), 21 visited Korea (K), and 6 visited neither country. (a) Explain why n(J ∪ K) = 34. (b) Find the number who visited both. (c) Find the number who visited exactly one country.

1. (a)Explain n(J ∪ K) = 34.[2]
2. (b)Find n(J ∩ K).[1]
3. (c)Find the “exactly one” count.[2]

Two sets satisfy n(A) = 2x, n(B) = x, n(A ∩ B) = 6 and n(A ∪ B) = 30. (a) Form an equation in x. (b) Solve it. (c) State n(A) and n(B).

1. (a)Form the equation.[2]
2. (b)Solve for x.[1]
3. (c)State n(A) and n(B).[2]

In a group of 50 people, 31 like tea, 24 like coffee, and everyone likes at least one of the two. (a) Find the number who like both. (b) Find the number who like tea only.

1. (a)Number who like both.[3]
2. (b)Number who like tea only.[2]

Let ξ = {integers from 1 to 20}, E = {even numbers} and M = {multiples of 4}. (a) Show that M ⊂ E. (b) State n(E ∪ M) with a reason. (c) List the elements that are in E but not in M.

1. (a)Show M ⊂ E.[2]
2. (b)n(E ∪ M) with reason.[2]
3. (c)List E ∩ M′.[2]

Sets A and B satisfy n(A ∩ B) = 7, n(A) = 15, n(B) = 13 and n(ξ) = 40. (a) Find n(A ∪ B). (b) How many elements are outside both sets? (c) Find n(A ∩ B′).

1. (a)n(A ∪ B).[2]
2. (b)Outside both.[2]
3. (c)n(A ∩ B′).[2]

In a class of 46 students, 6 pass neither Mathematics (M) nor Science (S). The number who pass M is twice the number who pass S, and 8 pass both. (a) Show that 3x − 8 = 40, where x students pass S. (b) Solve for x. (c) State n(M) and n(S).

1. (a)Show the equation.[3]
2. (b)Solve for x.[1]
3. (c)State n(M) and n(S).[2]

All 30 members of a club choose at least one of two activities, chess (C) or robotics (R). Given n(C) = 18 and n(R) = 15, find (a) n(C ∪ R), (b) n(C ∩ R), and (c) n(C′ ∩ R), the members who do robotics only.

1. (a)Find n(C ∪ R).[2]
2. (b)Find n(C ∩ R).[2]
3. (c)Find the robotics-only count.[2]