Venn diagrams and set notation

Evaluate — 2 marks

A physics laboratory classifies experiments based on two criteria: experiments that require a power supply (set P) and experiments that produce measurable electromagnetic radiation (set E). A technician creates a Venn diagram to organize 12 experiments. The diagram shows: 3 experiments in P only, 2 experiments in E only, 4 experiments in both P and E, and 3 experiments in neither set.

(a) Evaluate whether the technician's Venn diagram correctly represents all 12 experiments. Show your working. [1 mark]
(b) Using set notation, write an expression for the experiments that require a power supply but do NOT produce measurable electromagnetic radiation. [1 mark]

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(a) Adds the four regions: 3 + 2 + 4 + 3 = 12 ✓ OR states that all experiments are accounted for correctly
(a) Confirms the diagram is correct/valid
(b) Writes P ∩ E' or P − E or P \ E (or equivalent correct notation showing P only, excluding E)

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.

(a) Describe what the overlapping region in a Venn diagram represents when classifying laboratory equipment into 'measures length' and 'measures time'. [1 mark]
(b) Describe where a stopwatch would be placed in this Venn diagram and explain your answer. [1 mark]

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(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.

(a) Write down the numbers in A ∩ B. [1 mark]
(b) Show that n(A ∪ B) = 8. [2 marks]

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(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.

(a) One student is chosen at random from the class. Explain what the notation P(M ∪ E) represents in this context. [1 mark]
(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]

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(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}.

(a) State the elements of A ∩ B. [2 marks]
(b) State the value of n(A ∪ B). [2 marks]

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(a) 6
(a) 12
(b) 8

Evaluate — 2 marks

Describe — 2 marks

Show — 3 marks

Explain — 2 marks

State — 4 marks

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