Maths · Sets and functions · chapter 1 of 14

Class 11 Maths Chapter 1: Sets case study question with answers

One case-based question of the kind CBSE-pattern school exams ask: a passage, then 3 parts worth 4 marks in all, with the answer and the marking scheme for each. Written from the NCERT chapter.

Case study · 4 marks

The welfare committee of a housing society in Indore lists eight residents who signed up for weekend activities: U = {Aarav, Bina, Chetan, Divya, Esha, Farhan, Gopal, Hema}. The yoga group is Y = {Aarav, Bina, Divya, Farhan, Hema} and the cricket group is C = {Bina, Chetan, Farhan, Gopal}. The committee wants to know who is in both groups, who is in only one, and who has joined neither, so that it can plan the timings and send reminders to the right people.

Part 1 · 1 mark

Y ∩ C is

(a) {Farhan}(b) {Bina, Farhan}(c) {Bina, Chetan, Farhan}(d) {Aarav, Divya, Hema}

Show answer

Answer: (b) {Bina, Farhan}

Bina and Farhan are the only residents listed in both Y and C. Option (d) is Y − C.

Part 2 · 1 mark

The number of residents who are in exactly one of the two groups is

(a) 5(b) 3(c) 7(d) 6

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Answer: (a) 5

Y − C = {Aarav, Divya, Hema} and C − Y = {Chetan, Gopal}; these are disjoint, so 3 + 2 = 5 residents are in exactly one group.

Part 3 · 2 marks

Find (Y ∪ C)′ and Y′ ∩ C′, and say what the answer means for the committee.

Answer and marking scheme

Y ∪ C = {Aarav, Bina, Chetan, Divya, Farhan, Gopal, Hema}, so (Y ∪ C)′ = {Esha}. Y′ = {Chetan, Esha, Gopal}, C′ = {Aarav, Divya, Esha, Hema}, so Y′ ∩ C′ = {Esha}. The two are equal, as De Morgan's law says: Esha is the only resident who has joined neither group.

(Y ∪ C)′ = {Esha} (1)
Y′ ∩ C′ = {Esha} and the interpretation (1)

Find (Y ∩ C)′ and Y′ ∪ C′, and say what the answer means for the committee.

Answer and marking scheme

(Y ∩ C)′ = U − {Bina, Farhan} = {Aarav, Chetan, Divya, Esha, Gopal, Hema}. Y′ ∪ C′ = {Chetan, Esha, Gopal} ∪ {Aarav, Divya, Esha, Hema} = {Aarav, Chetan, Divya, Esha, Gopal, Hema}. The two are equal, as De Morgan's law says: these six residents are not in both groups, so their timings do not clash.

(Y ∩ C)′ found (1)
Y′ ∪ C′ found, equal, with the interpretation (1)

Chapter 2: Relations and Functions →