Describe — 2 marks
A physics teacher is organising a set of laboratory equipment. Some items can be used to measure length, some can be used to measure time, and some can be used for both purposes. The teacher decides to use a Venn diagram to classify the equipment.
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(a) Describe what the overlapping region in a Venn diagram represents when classifying laboratory equipment into 'measures length' and 'measures time'.
[1 mark]
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(b) Describe where a stopwatch would be placed in this Venn diagram and explain your answer.
[1 mark]
Show mark scheme
- (a) The overlapping region represents equipment that can measure both length and time / equipment that belongs to both sets
- (b) A stopwatch would be placed in the 'measures time' region only / outside the 'measures length' region, because a stopwatch measures time but not length
Show — 3 marks
A teacher asks students to consider numbers from 1 to 12.
Set A = {even numbers}
Set B = {multiples of 3}
The Venn diagram shows this information.
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(a) Write down the numbers in A ∩ B.
[1 mark]
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(b) Show that n(A ∪ B) = 8.
[2 marks]
Show mark scheme
- (a) 6, 12 (both required for the mark)
- (b) Method: n(A) + n(B) − n(A ∩ B) or lists all elements of A ∪ B
- (b) Correct completion showing 6 + 4 − 2 = 8 or listing {2, 3, 4, 6, 8, 9, 10, 12}
Explain — 2 marks
A teacher surveys 30 students about their favourite subjects. The Venn diagram shows the sets: M = students who like Mathematics and E = students who like English. The numbers in each region represent the number of students.
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(a) One student is chosen at random from the class. Explain what the notation P(M ∪ E) represents in this context.
[1 mark]
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(b) There are 12 students who like Mathematics, 15 who like English, and 7 who like both. Explain why the number of students who like neither subject is 10.
[1 mark]
Show mark scheme
- (a) The probability of selecting a student who likes Mathematics OR English (or both)
- (b) 12 + 15 − 7 = 20 students like at least one subject, so 30 − 20 = 10 like neither
State — 4 marks
The Venn diagram shows two sets, A and B, within the universal set ξ. Set A contains the even numbers from 2 to 12, and set B contains the multiples of 3 from 3 to 12. The universal set ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.
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(a) State the elements of A ∩ B.
[2 marks]
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(b) State the value of n(A ∪ B).
[2 marks]
Show mark scheme
Compare — 3 marks
The Venn diagram shows the universal set ε = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20}. Set A contains the even numbers in ε. Set B contains the multiples of 3 in ε.
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(a) List the elements of A ∩ B.
[1 mark]
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(b) Compare n(A ∩ B') with n(A' ∩ B). Show your working.
[2 marks]
Show mark scheme
- (a) {6, 12, 18} (accept without brackets, any order)
- (b) Correctly identifies A ∩ B' = {2, 4, 8, 10, 14, 16, 20} so n(A ∩ B') = 7
- (b) Correctly identifies A' ∩ B = {3, 9, 15} so n(A' ∩ B) = 3 AND states that n(A ∩ B') > n(A' ∩ B) (or equivalent comparison, e.g., 7 > 3)
GCSE Perfect