Sets Miscellaneous Exercise 01
Question:
In a group of students, 100 students know Hindi,50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group?
Answer:
Answer: There are 175 students in the group.
Question:
In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only
Answer:

Total number of people surveyed = 21 + 26 + 29 = 76

Number of people who liked product C only = 29  (14 + 12 + 8) = 29  34 = 5
Question:
Examine whether the following statements are true or false: (i) { a,b } ⊄ { b,c,a } (ii) { a,e } ⊂ { x:x is a vowel in the English alphabet} (iii) { 1,2,3 } ⊂ { 1,3,5 } (iv) { a }⊂ { a,b,c } (v) { a }ϵ { a,b,c } (vi) { x:x is an even natural number less than 6} ⊂ { x : x is a natural number which divides 36 }
Answer:
(i) True (ii) True (iii) False (iv) True (v) True (vi) True
Question:
Let A=(1,2,(3,4),5). Which of the following statements are incorrect and why? (i) {3, 4}⊂A (ii) {3, 4}∈A (iii) {{3, 4}}⊂A (iv) 1∈A (v) 1⊂A (vi) {1, 2, 5}⊂A (vii) {1, 2, 5}∈A (viii) {1, 2, 3}⊂A (ix) ϕ∈A (x) ϕ⊂A (xi) {ϕ}⊂A
Answer:
(i) {3, 4}⊂A  Incorrect. A cannot be multiplied by a tuple. (ii) {3, 4}∈A  Correct. (3,4) is an element of A. (iii) {{3, 4}}⊂A  Incorrect. A cannot be multiplied by a tuple. (iv) 1∈A  Incorrect. A cannot be multiplied by a scalar. (v) 1⊂A  Incorrect. A cannot be multiplied by a scalar. (vi) {1, 2, 5}⊂A  Incorrect. A cannot be multiplied by a tuple. (vii) {1, 2, 5}∈A  Incorrect. A cannot be multiplied by a tuple. (viii) {1, 2, 3}⊂A  Incorrect. A cannot be multiplied by a tuple. (ix) ϕ∈A  Incorrect. A cannot be multiplied by a scalar. (x) ϕ⊂A  Incorrect. A cannot be multiplied by a scalar. (xi) {ϕ}⊂A  Incorrect. A cannot be multiplied by a tuple.
Question:
Given the sets A={1, 3, 5}, B={2, 4, 6} and C={0, 2, 4, 6, 8}, which of the following may be considered as universal set (s) for all the three sets A, B and C i){0, 1, 2, 3, 4, 5, 6} ii)ϕ iii){0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} iv){0, 1, 2, 3, 4, 5, 6, 7, 8}
Answer:
i) {0, 1, 2, 3, 4, 5, 6} This is a valid universal set for all the three sets A, B and C.
Question:
Find sets A, B and C such that A∩B,B∩C and A∩C are nonempty sets and A∩B∩C=ϕ
Answer:
A = {1, 2, 3, 4, 5} B = {2, 4, 6, 8, 10} C = {3, 6, 9, 12, 15}
A ∩ B = {2, 4} B ∩ C = {6} A ∩ C = {3} A ∩ B ∩ C = ϕ
Question:
In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find: (i) the number of people who read at least one of the newspapers. (ii) the number of people who read exactly one newspaper.
Answer:
(i) 60 people
(ii) 20 people
Question:
How many elements has P(A) , if A=ϕ?
Answer:
Answer: P(A) = {ϕ}
Therefore, P(A) has one element, namely ϕ.
Question:
Make correct statements by filling in the symbols ⊂ or ⊄ in blank spaces: (i) {2,3,4}……{1,2,3,4,5} (ii) {a,b,c}…{b,c,d} (iii) {x:x is a student of Class XI of your school}. . .{x:x student of your school} (iv) {x:x is a circle in the plane} . . .{x:x is a circle in the same plane with radius 1 unit} (v) {x:x is a triangle in a plane} . . . {x:x is a rectangle in the plane} (vi) {x:x is an equilateral triangle in a plane} . . . {x:x is a triangle in the same plane} (vii) {x:x is an even natural number} . . . {x:x is an integer}
Answer:
(i) ⊂ (ii) ⊂ (iii) ⊂ (iv) ⊄ (v) ⊄ (vi) ⊂ (vii) ⊂
Question:
Show that A∩B=A∩C need not imply B=C
Answer:

Assume A∩B=A∩C

This implies that all elements in A∩B are also in A∩C

However, this does not necessarily mean that all elements in B are also in C, or vice versa.

Therefore, A∩B=A∩C does not necessarily imply B=C.
Question:
Write the following as intervals : (i) {x:xϵR,4<x≤6} (ii) {x:xϵR,−12<x<−10} (iii) {x:xϵR,0≤x<7} (iv) {x:xϵR,3≤x≤4}
Answer:
(i) (4, 6] (ii) (12, 10) (iii) [0, 7) (iv) [3, 4]
Question:
In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many students were taking neither tea nor coffee?
Answer:
Step 1: Add the number of students taking tea and coffee together. Answer: 150 + 225 = 375
Step 2: Subtract the number of students taking both tea and coffee from the total. Answer: 375  100 = 275
Step 3: Subtract the number of students taking tea and coffee from the total number of students surveyed. Answer: 600  275 = 325
Therefore, the number of students taking neither tea nor coffee is 325.
Question:
Let A and B be sets. If A∩X=B∩X=ϕ and A∪X=B∪X for some set X, show that A=B
Answer:
 Assume A≠B.
 Since A∩X=B∩X=ϕ, A and B have no elements in common with X.
 Since A∪X=B∪X, A and B must have the same elements in X.
 This contradicts the assumption that A≠B, thus A=B.
Question:
Write down all the subsets of the following sets (i) {a} (ii) {a, b} (iii) {1, 2, 3} (iv) ϕ
Answer:
(i) {a}, ϕ (ii) {a, b}, {a}, {b}, ϕ (iii) {1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ϕ (iv) ϕ
Question:
What universal set(s) would you propose for each of the following: (i) The set of right triangles. (ii) The set of isosceles triangles
Answer:
(i) Universal set: U = {all triangles}
(ii) Universal set: U = {all triangles}
Question:
Write the following intervals in setbuilder form: (i) (−3,0) (ii) [6,12] (iii) (6,12] (iv) [23,5)
Answer:
(i) {x  3 < x < 0}
(ii) {x  6 ≤ x ≤ 12}
(iii) {x  6 < x ≤ 12}
(iv) {x  23 ≤ x ≤ 5}