SATHEE: Sets Exercises 01

Sets Exercises 01

Question:

Are the following pair of sets equal ? Give reasons. (i) A={2,3},B={x:x is solution of x²+5x+6=0} (ii) A={x:x is a letter in the word FOLLOW} B={y:y is a letter in the word WOLF}

Answer:

(i) No, the sets are not equal. Set A contains the numbers 2 and 3, while set B contains the solutions of the equation x²+5x+6=0, which are -2 and -3.

(ii) Yes, the sets are equal. Set A contains the letters F, O, L, L, O, and W, while set B contains the letters W, O, L, and F, which are the same letters.

Question:

Which of the following pairs of sets are disjoint (i) {1,2,3,4} and {x:x is a natural number and 4≤x≤6} (ii) { a,e,i,o,u } and { c,d,e,f } (iii) {x:x is an even integer } and {x:x is an odd integer}

Answer:

(i) The sets {1,2,3,4} and {x:x is a natural number and 4≤x≤6} are not disjoint because the set {x:x is a natural number and 4≤x≤6} contains the elements 5 and 6 which are also in the set {1,2,3,4}.

(ii) The sets { a,e,i,o,u } and { c,d,e,f } are disjoint because there are no elements that are shared between the two sets.

(iii) The sets {x:x is an even integer} and {x:x is an odd integer} are disjoint because there are no elements that are shared between the two sets.

Question:

Let U={1,2,3,4,5,6,7,8,9},A={1,2,3,4},B={2,4,6,8} and C={3,4,5,6}. Find (i) A′ (ii) B′ (iii) (A∪C)′ (iv) (A∪B)′ (v) (A′)′ (vi) (B−C)′

Answer:

(i) A’ = {5,7,8,9} (ii) B’ = {1,3,5,7,9} (iii) (A∪C)’ = {2,7,8,9} (iv) (A∪B)’ = {5,7,9} (v) (A’)’ = {1,2,3,4} (vi) (B−C)’ = {2,8}

Question:

Which of the following are examples of the null set? (i) Set of odd natural numbers divisible by 2 (ii) Set of even prime numbers (iii) {x:x is a natural numbers, x < 5 and x > 7} (iv) {y:y is a point common to any two parallel lines}

Answer:

(i) Set of odd natural numbers divisible by 2 - Null set (ii) Set of even prime numbers - Null set (iii) {x:x is a natural numbers, x < 5 and x > 7} - Null set (iv) {y:y is a point common to any two parallel lines} - Not a null set

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) False (iii) False (iv) True (v) True (vi) False

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} may be considered as a universal set for all the three sets A, B and C.

ii) ϕ cannot be considered as a universal set for all the three sets A, B and C.

iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} may be considered as a universal set for all the three sets A, B and C.

iv) {0, 1, 2, 3, 4, 5, 6, 7, 8} may be considered as a universal set for all the three sets A, B and C.

Question:

Find the union of each of the following pairs of sets: (i) X={1,3,5}Y={1,2,3} (ii) A={a,e,i,o,u},B={a,b,c} (iii) A={x:x is a natural number and multiple of 3}. B={x:x is a natural number less than 6}. (iv) A={x:x is a natural number and 1<x≤6} B={x:x is a natural number and6<x<10}. (v) A={1,2,3}, B=ϕ

Answer:

(i) X ∪ Y = {1,2,3,5}

(ii) A ∪ B = {a,b,c,e,i,o,u}

(iii) A ∪ B = {x:x is a natural number and multiple of 3 or less than 6}

(iv) A ∪ B = {x:x is a natural number and 1<x≤10}

(v) A ∪ B = {1,2,3}

Question:

If R is the set of real numbers and Q is the set of rational numbers, then what is R−Q?

Answer:

Answer: R−Q is the set of real numbers that are not rational numbers, also known as irrational numbers.

Question:

If A={x:x is a natural number },B={x:x is an even natural number}C={x:x is an odd natural number} and D={x:x is a prime number }, find (i) A∩B (ii) A∩C (iii) A∩D (iv) B∩C (v) B∩D (vi) C∩D

Answer:

(i) A∩B = {x:x is an even natural number}

(ii) A∩C = {x:x is an odd natural number}

(iii) A∩D = {x: x is a prime number and a natural number}

(iv) B∩C = ∅

(v) B∩D = {x: x is a prime even number}

(vi) C∩D = {x: x is a prime odd number}

Question:

Which of the following sets are finite or infinite (i) The set of months of a year (ii) {1,2,3,….} (iii) {1,2,3,….99,100} (iv) The set of positive integers greater than 100 (v) The set of prime numbers less than 99

Answer:

(i) The set of months of a year: Finite (ii) {1,2,3,….}: Infinite (iii) {1,2,3,….99,100}: Finite (iv) The set of positive integers greater than 100: Infinite (v) The set of prime numbers less than 99: Finite

Question:

Let U={1,2,3,4,5,6,7,8,9},A={1,2,3,4},B={2,4,6,8} and C={3,4,5,6}. Find (i) A′ (ii) B′ (iii) (A∪C)′ (iv) (A∪B)′ (v) (A′)′ (vi) (B−C)′

Answer:

(i) A′ = {5,6,7,8,9}

(ii) B′ = {1,3,5,7,9}

(iii) (A∪C)′ = {7,8,9}

(iv) (A∪B)′ = {5,7,9}

(v) (A′)′ = A = {1,2,3,4}

(vi) (B−C)′ = {1,2,7,8,9}

Question:

Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60o, what is A'?

Answer:

A = {all triangles in the plane that have at least one angle different from 60°}

Question:

How many elements has P(A) , if A=ϕ?

Answer:

Answer: P(A) has 0 elements, since A=ϕ (the empty set).

Question:

Let A={a,b},B={a,b,c}. Is A⊂B? What is A∪B?

Answer:

A⊂B: Yes, A is a subset of B.

A∪B = {a, b, c}

Question:

If X={a,b,c,d} and Y={f,b,d,g}, find (i) X−Y (ii) Y−X (iii) X∩Y

Answer:

(i) X−Y = {a, c}

(ii) Y−X = {f, g}

(iii) X∩Y = {b, d}

Question:

If U={1,2,3,4,5,6,7,8,9},A={2,4,6,8} and B={2,3,5,7}. Verify that (i) (A∪B)′=A′∩B′ (ii) (A∩B)′=A′∪B′

Answer:

(i) (A∪B)' = (A’∩B')

A’ = {1,3,5,7,9} B’ = {1,4,6,9}

A’∩B’ = {1,9}

Hence, (A∪B)’ = {1,9}

(ii) (A∩B)' = (A’∪B')

A’ = {1,3,5,7,9} B’ = {1,4,6,9}

A’∪B’ = {1,3,4,5,6,7,9}

Hence, (A∩B)’ = {1,3,4,5,6,7,9}

Question:

In a committee, 50 people speak French, 20 speak Spanish and 10 speak both Spanish and French. How many speak at least one of these two languages?

Answer:

50 people speak French.
20 people speak Spanish.
10 people speak both Spanish and French.
Therefore, the total number of people who speak at least one of these two languages is 80 (50 + 20 + 10).

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:

Fill in the blanks to make each of the following a true statement: (i) A∪A′=… (ii) ϕ′∩A=… (iii) A∩A′=… (iv) U′∩A=…

Answer:

(i) A∪A′=U (ii) ϕ′∩A=A (iii) A∩A′=ϕ (iv) U′∩A=ϕ

Question:

In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?

Answer:

First, count the number of people who like coffee: 37
Then, count the number of people who like tea: 52
Subtract the number of people who like coffee from the total number of people in the group: 70 - 37 = 33
Subtract the number of people who like tea from the total number of people in the group: 70 - 52 = 18
Add the two numbers together: 33 + 18 = 51
The answer is 51 people like both coffee and tea.

Question:

Taking the set of natural numbers as the universal set, write down the complements of the following sets: (i) {x:x is an even natural number} (ii) {x:x is an odd natural number} (iii) {x:x is a positive multiple of 3} (iv) { x:x is a prime number } (v) {x:x is a natural number divisible by 3 and 5} (vi) {x:x is a perfect square } (vii) { x:x is a perfect cube} (viii) {x:x+5=8} (ix) {x:2x+5=9} (x) {x:x≥7} (xi) {x:xϵN and 2x+1>10}

Answer:

(i) {x:x is an odd natural number}

(ii) {x:x is an even natural number}

(iii) {x:x is a positive multiple of 3 that is not divisible by 5}

(iv) {x:x is a composite number}

(v) {x:x is a natural number not divisible by 3 and 5}

(vi) {x:x is a positive natural number that is not a perfect square}

(vii) {x:x is a positive natural number that is not a perfect cube}

(viii) {x:x+5≠8}

(ix) {x:2x+5≠9}

(x) {x:x<7}

(xi) {x:xϵN and 2x+1≤10}

Question:

If X and Y are two sets such that n(X)=17,n(Y)=23 and n(X∪Y)=38, find n(X∩Y)

Answer:

Step 1: n(X∩Y) = n(X) + n(Y) - n(X∪Y)

Step 2: n(X∩Y) = 17 + 23 - 38

Step 3: n(X∩Y) = 12

Question:

In a group of 400 people, 250 can speak Hindi and 200 can speak English. How many people can speak both Hindi and English?

Answer:

First, count the total number of people in the group: 400
Then, count the number of people who can speak Hindi: 250
Next, count the number of people who can speak English: 200
Finally, subtract the total number of people (400) from the sum of the number of people who can speak Hindi (250) and English (200): 400 - (250 + 200) = -50
The answer is 0, because it is impossible for a negative number of people to speak both Hindi and English.

Question:

If S and T are two sets such that S has 21 elements, T has 32 elements, and S∩T has 11 elements, how many elements does S∪T have?

Answer:

Answer: S∪T has 43 elements.

Question:

In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?

Answer:

Number of people who like tennis only and not cricket: 15
Number of people who like tennis: 25 (15 + 10)

Question:

In the following, state whether A=B or not: (i) A={a,b,c,d}B={d,c,b,a} (ii) A={4,8,12,16}B={8,4,16,18} (iii) A={2,4,6,8,10}B={x:x is a positive even integer and x≤10} (iv) A={x:x is a multiple of 10},B={10,15,20,25,30,…}

Answer:

(i) A=B (ii) A≠B (iii) A=B (iv) A=B

Question:

From the sets given below, select equal set: A={2,4,8,12}, B={1,2,3,4}, C={4,8,12,14}, D={3,1,4,2}, E={−1,1}, F={0,a}, G={−1,1}, H={0,1}

Answer:

Answer: E and G are equal sets, as they both contain the elements {-1, 1}.

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:

If A={1,2,3,4},B={3,4,5,6},C={5,6,7,8} and D={7,8,9,10}; find (i) A∪B (ii) A∪C (iii) B∪C (iv) B∪D (v) A∪B∪C (vi) A∪B∪D (vii) B∪C∪D

Answer:

(i) A∪B = {1,2,3,4,5,6} (ii) A∪C = {1,2,3,4,5,6,7,8} (iii) B∪C = {3,4,5,6,7,8} (iv) B∪D = {3,4,5,6,7,8,9,10} (v) A∪B∪C = {1,2,3,4,5,6,7,8} (vi) A∪B∪D = {1,2,3,4,5,6,7,8,9,10} (vii) B∪C∪D = {3,4,5,6,7,8,9,10}

Question:

Find the intersection of each pair of sets: (i) X={1,3,5} Y={1,2,3} (ii) A={a,e,i,o,u} B={a,b,c} (iii) A={x:x is a natural number and multiple of 3} B={x:x is a natural number less than 6} (iv) A={x:x is a natural number and 1<x≤6} B={x:x is a natural number and 6 < x < 10} (v) A={1,2,3}, B=ϕ

Answer:

(i) X∩Y = {1,3}

(ii) A∩B = {a}

(iii) A∩B = {3}

(iv) A∩B = ϕ

(v) A∩B = ϕ

Question:

If A and B are two sets such that A⊂B, then what is A∪B?

Answer:

A∪B = B

Explanation: The union of two sets A and B is the set of all elements that are in either A or B. Since A is a subset of B, all elements of A are also in B. Therefore, A∪B = B.

Question:

Let A={1,2,{3,4},5}. Which of the following statements are incorrect and why? (i) {3,4}⊂A (ii) {3,4}ϵA (iii) {{3,4}}⊂A (iv) 1ϵA (v) 1⊂A (vi) {1,2,5}⊂A (vii) {1,2,5}ϵA (viii) {1,2,3}⊂A (ix) ϕϵA (x) ϕ⊂A (xi) {ϕ}⊂A

Answer:

(i) Correct (ii) Correct (iii) Incorrect - {3,4} is an element of A, not a subset of A. (iv) Correct (v) Incorrect - 1 is an element of A, not a subset of A. (vi) Correct (vii) Incorrect - {1,2,5} is a subset of A, not an element of A. (viii) Incorrect - {1,2,3} is not a subset of A. (ix) Incorrect - ϕ (the empty set) is not an element of A. (x) Incorrect - ϕ is not a subset of A. (xi) Incorrect - ϕ is not a subset of A.

Question:

Write down all the subsets of the following sets (i) {a} (ii) {a, b} (iii) {1, 2, 3} (iv) ϕ

Answer:

(i) {a}

(ii) {a}, {b}, {a,b}

(iii) {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}

(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:

If A={3,5,7,9,11},B={7,9,11,13},C={11,13,15} and D={15,17};find (i) A∩B (ii) B∩C (iii) A∩C∩D (iv) A∩C (v) B∩D (vi) A∩(B∪C) (vii) A∩D (viii) A∩(B∪D) (ix) (A∩B)∩(B∩C) (x) (A∪D)∩(B∪C)

Answer:

(i) A∩B = {7,9,11}

(ii) B∩C = {11,13}

(iii) A∩C∩D = {11}

(iv) A∩C = {11}

(v) B∩D = {15}

(vi) A∩(B∪C) = {3,5,7,9,11}

(vii) A∩D = ∅

(viii) A∩(B∪D) = {3,5,7,9,11,15}

(ix) (A∩B)∩(B∩C) = {11}

(x) (A∪D)∩(B∪C) = {3,5,7,9,11,13,15}

Question:

If A={3,6,9,12,15,18,21},B={4,8,12,16,20}, C={2,4,6,8,10,12,14,16},D={5,10,15,20}; find (i) A−B (ii) A−C (iii) A−D (iv) B−A (v) C−A (vi) D−A (vii) B−C (viii) B−D (ix) C−B (x) D−B (xi) C−D (xii) D−C

Answer:

(i) A−B = {3,6,9,15,18,21}

(ii) A−C = {9,12,15,18,21}

(iii) A−D = {3,6,9,12,15,18,21}

(iv) B−A = {4,8,16,20}

(v) C−A = {2,4,6,8,10,14,16}

(vi) D−A = {5,10,15,20}

(vii) B−C = {4,8,12,16,20}

(viii) B−D = {4,8,12,16}

(ix) C−B = {2,6,10,14,16}

(x) D−B = {5,10,15,20}

(xi) C−D = {2,4,6,8,10,14,16}

(xii) D−C = {5,10,15,20}

Question:

Draw appropriate Venn diagram for each of the following: (i) (A∪B)′, (ii) A′∩B′, (iii) (A∩B)′, (iv) A′∪B′

Answer:

(i) A B

(ii) A’ B'

(iii) A B

(iv) A’ B'

Question:

If X and Y are two sets such that X∪Y has 18 elements, X has 8 elements and Y has 15 elements; how many elements does X∩Y have?

Answer:

X∪Y has 18 elements.
X has 8 elements.
Y has 15 elements.
X∩Y has 8 - (18 - 15) = 1 element.

Question:

Write the following intervals in set-builder form: (i) (−3,0) (ii) [6,12] (iii) (6,12] (iv) [23,5)

Answer:

(i) {x | x > -3 and x < 0}

(ii) {x | x ≥ 6 and x ≤ 12}

(iii) {x | x > 6 and x ≤ 12}

(iv) {x | x ≥ 23 and x ≤ 5}

Question:

State whether each of the following statement is true or false. Justify your answer. (i) {2,3,4,5} and {3,6} are disjoint sets. (ii) {a,e,i,o,u} and {a,b,c,d} are disjoint sets (iii) {2,6,10,14} and {3,7,11,15} are disjoint sets (iv) {2,6,10} and {3,7,11} are disjoint sets

Answer:

(i) False. {2,3,4,5} and {3,6} are not disjoint sets because they share the element 3.

(ii) False. {a,e,i,o,u} and {a,b,c,d} are not disjoint sets because they share the element a.

(iii) True. {2,6,10,14} and {3,7,11,15} are disjoint sets because they do not share any elements.

(iv) False. {2,6,10} and {3,7,11} are not disjoint sets because they share the element 6.

Question:

If U={a,b,c,d,e,f,g,h}, find the complements of the following sets: (i) A={a,b,c} (ii) B={d,e,f,g} (iii) C={a,c,e,g} (iv) D={f,g,h,a}

Answer:

(i) A’ = {d,e,f,g,h} (ii) B’ = {a,b,c} (iii) C’ = {b,d,f,h} (iv) D’ = {b,c,e}

Question:

If X and Y are two sets such that X has 40 elements, X∪Y has 60 elements and X∩Y has 10 elements, how many elements does Y have?

Answer:

Y has 50 elements.

Explanation: We know that X has 40 elements and X∪Y has 60 elements. This means that the total number of elements in X and Y combined is 60. We also know that X∩Y has 10 elements. This means that the number of elements that are common between X and Y is 10.

Therefore, the number of elements in Y must be equal to the total number of elements in X and Y (60) minus the number of elements that are common between X and Y (10).

60 - 10 = 50

Therefore, Y has 50 elements.

Question:

State whether each of the following set is finite or infinite: (i) The set of lines which are parallel to the x-axis (ii) The set of letters in the English alphabet (iii) The set of numbers which are multiple of 5 (iv) The set of animals living on the earth (v) The set of circles passing through the origin (0,0)

Answer:

(i) Infinite (ii) Finite (iii) Infinite (iv) Infinite (v) Infinite