State whether or not the following elements form a set; if not, give reason:
(i) All the easy problems in your text book.
(ii) All the three-sided figures.
(iii) The first five whole numbers.
(iv) All the tall boys of your class.
(v) The last three days of a week.
(vi) All the triangles that are difficult to draw.
(vii) The first three letters of the English alphabet.
(viii) All the tasty fruits.
(ix) All the clever boys of class 6.
(x) All the good schools in Delhi.
(xi) All the girls in your class whose heights are less than your height.
(xii) All the boys in your class whose heights are more than your height.
(xiii) All the problems in your Mathematics book that are difficult for Amit.
Answer
A set is a collection of well-defined objects.
(i) It is not a set; some problems may be easy for one person but may not be easy to some other person. So, the objects are not well defined.
(ii) It is a set; the three-sided figures (triangles) are well defined.
(iii) It is a set; the first five whole numbers, i.e. 0, 1, 2, 3 and 4, are well defined.
(iv) It is not a set; it is not mentioned that the boys must be taller than which boy. If we consider three boys A, B and C, boy A can be taller than B but not necessarily taller than C. So, the objects are not well defined.
(v) It is a set; the last three days of a week, i.e. Friday, Saturday and Sunday, are well defined.
(vi) It is not a set; it may be difficult for one student to draw a given triangle, but to some other student it may not be difficult to draw the same triangle. So, the objects are not well defined.
(vii) It is a set; the first three letters of the English alphabet, i.e. A, B and C, are well defined.
(viii) It is not a set; a fruit may be tasty for one person and may not be tasty to other person/persons. So, the objects are not well defined.
(ix) It is not a set; clever in what respect? The word 'clever' is not well defined.
(x) It is not a set; all the people cannot find the same schools as good schools. So, the objects are not well defined.
(xi) It is a set; the heights are compared with a fixed height (your height), so it can be clearly decided which girls are to be included. The objects are well defined.
(xii) It is a set; the heights are compared with a fixed height (your height), so it can be clearly decided which boys are to be included. The objects are well defined.
(xiii) It is a set; it can be clearly found which problems are difficult for Amit and which are not. So, the objects are well defined.
If set A = { 2, 3, 4, 5, 6 }, state which of the following statements are true and which are false:
(i) 2 ∈ A
(ii) 5, 6 ∈ A
(iii) 3, 4, 7 ∈ A
(iv) 2, 8 ∈ A
Answer
A = { 2, 3, 4, 5, 6 }
(i) 2 is a member of A.
True
(ii) Both 5 and 6 are members of A.
True
(iii) 3 and 4 are members of A, but 7 is not a member of A.
False
(iv) 2 is a member of A, but 8 is not a member of A.
False
If set B = { 4, 6, 8, 10, 12, 14 }, state which of the following statements are correct and which are wrong:
(i) 5 ∈ B
(ii) 12 ∈ B
(iii) 14 ∈ B
(iv) 9 ∈ B
(v) B is the set of even numbers between 2 and 16.
(vi) 4, 6 and 10 are the members of the set B.
Also, write the wrong statements correctly.
Answer
B = { 4, 6, 8, 10, 12, 14 }
(i) 5 is not a member of B.
Wrong; the correct statement is 5 ∉ B.
(ii) 12 is a member of B.
Correct
(iii) 14 is a member of B.
Correct
(iv) 9 is not a member of B.
Wrong; the correct statement is 9 ∉ B.
(v) The even numbers between 2 and 16 are 4, 6, 8, 10, 12 and 14, which are exactly the elements of B.
Correct
(vi) 4, 6 and 10 are all members of B.
Correct
State whether true or false:
(i) Sets { 4, 9, 6, 2 } and { 6, 2, 4, 9 } are not the same.
(ii) Sets { 0, 1, 3, 9, 4 } and { 4, 0, 1, 3, 9 } are the same.
(iii) Sets { 5, 4 } and { 5, 4, 4, 5 } are not the same.
(iv) Sets { 8, 3 } and { 3, 3, 8 } are the same.
(v) Collection of vowels used in the word 'ALLAHABAD' forms a set.
(vi) If P is the set of letters in the word 'ROOP'; then P = { p, o, r }
(vii) If M is the set of letters used in the word 'MUMBAI', then: M = { m, u, b, a, i }
Answer
(i) The two sets have the same elements 4, 9, 6 and 2. A change in order does not change a set, so they are the same. Hence, the statement that they are not the same is
False
(ii) Both sets have the same elements 0, 1, 3, 9 and 4. So, they are the same.
True
(iii) Repetition of elements does not change a set, so { 5, 4 } and { 5, 4, 4, 5 } are the same. Hence, the statement that they are not the same is
False
(iv) Repetition of elements does not change a set, so { 8, 3 } and { 3, 3, 8 } are the same.
True
(v) The collection of vowels in 'ALLAHABAD' is well defined (the only vowel used is A). So, it forms a set.
True
(vi) The letters in the word 'ROOP' are R, O, O, P. Writing each letter only once, the set of letters is { r, o, p } = { p, o, r }.
True
(vii) The letters in the word 'MUMBAI' are M, U, M, B, A, I. Writing each letter only once, M = { m, u, b, a, i }.
True
Write the set containing:
(i) the first five natural numbers.
(ii) the three types of angles.
(iii) the three types of triangles based on the length of their sides.
(iv) the members of your family.
(v) the first six consonants of the English alphabet.
(vi) the first four vowels of the English alphabet.
(vii) the names of any three Prime Ministers of India.
Answer
(i) The first five natural numbers are 1, 2, 3, 4 and 5.
The required set = { 1, 2, 3, 4, 5 }
(ii) The three types of angles (based on measure) are acute angle, right angle and obtuse angle.
The required set = { acute angle, right angle, obtuse angle }
(iii) The three types of triangles based on the length of their sides are scalene triangle, isosceles triangle and equilateral triangle.
The required set = { scalene triangle, isosceles triangle, equilateral triangle }
(iv) The members of your family form a well-defined collection.
The required set = { Write the name of each member of your family }
(v) The first six consonants of the English alphabet are b, c, d, f, g and h.
The required set = { b, c, d, f, g, h }
(vi) The first four vowels of the English alphabet are a, e, i and o.
The required set = { a, e, i, o }
(vii) Three Prime Ministers of India are Indira Gandhi, Atal Bihari Vajpayee and Dr. Manmohan Singh.
The required set = { Indira Gandhi, Atal Bihari Vajpayee, Dr. Manmohan Singh }
(a) Write the members (elements) of each set given below:
(i) { 3, 8, 5, 15, 12, 7 }
(ii) { c, m, n, o, s }
(b) Write the sets whose elements are:
(i) 2, 4, 8, 16, 64 and 128
(ii) 3, 5, 15, 45, 75 and 90
Answer
(a)
(i) The members of the set { 3, 8, 5, 15, 12, 7 } are
3, 8, 5, 15, 12 and 7.
(ii) The members of the set { c, m, n, o, s } are
c, m, n, o and s.
(b)
(i) The set whose elements are 2, 4, 8, 16, 64 and 128 is
{ 2, 4, 8, 16, 64, 128 }
(ii) The set whose elements are 3, 5, 15, 45, 75 and 90 is
{ 3, 5, 15, 45, 75, 90 }
(i) Write the set of letters used in the word 'BHOPAL'.
(ii) Write the set of vowels used in the word 'BENGAL'.
(iii) Write the set of consonants used in the word 'HONG KONG'.
Answer
(i) The letters used in the word 'BHOPAL' are B, H, O, P, A, L.
The required set = { b, h, o, p, a, l }
(ii) The vowels used in the word 'BENGAL' are E and A.
The required set = { e, a }
(iii) The letters in the word 'HONG KONG' are H, O, N, G, K, O, N, G. The consonants among these are H, N, G, K, N, G. Writing each consonant only once, we get H, N, G, K.
The required set = { h, n, g, k }
Write each of the following sets in the Roster Form:
(i) The set of five numbers each of which is divisible by 3.
(ii) The set of integers between -4 and 4.
(iii) { x : x is a letter in the word 'SCHOOL' }
(iv) { x : x is an odd natural number between 10 and 20 }
(v) { Vowels used in the word 'AMERICA' }
(vi) { Consonants used in the word 'MADRAS' }
Answer
(i) Five numbers each divisible by 3 are 3, 6, 9, 12 and 15.
The required set = { 3, 6, 9, 12, 15 }
(ii) The integers between -4 and 4 are -3, -2, -1, 0, 1, 2 and 3.
The required set = { -3, -2, -1, 0, 1, 2, 3 }
(iii) The letters in the word 'SCHOOL' are S, C, H, O, O, L. Writing each letter only once, we get S, C, H, O, L.
{ x : x is a letter in the word 'SCHOOL' } = { s, c, h, o, l }
(iv) The odd natural numbers between 10 and 20 are 11, 13, 15, 17 and 19.
{ x : x is an odd natural number between 10 and 20 } = { 11, 13, 15, 17, 19 }
(v) The vowels used in the word 'AMERICA' are A, E, I, A. Writing each vowel only once, we get A, E, I.
{ Vowels used in the word 'AMERICA' } = { a, e, i }
(vi) The consonants used in the word 'MADRAS' are M, D, R, S.
{ Consonants used in the word 'MADRAS' } = { m, d, r, s }
Write each given set in the Roster Form:
(i) All prime numbers between 1 and 20.
(ii) The squares of the first four natural numbers.
(iii) Even numbers between 1 and 9.
(iv) The first eight letters of the English alphabet.
(v) The letters of the word 'BASKET'.
(vi) Four cities of India whose names start with the letter J.
(vii) Any four closed geometrical figures.
(viii) Vowels used in the word 'MONDAY'.
(ix) Single digit numbers that are perfect squares as well.
Answer
(i) The prime numbers between 1 and 20 are 2, 3, 5, 7, 11, 13, 17 and 19.
The required set = { 2, 3, 5, 7, 11, 13, 17, 19 }
(ii) The first four natural numbers are 1, 2, 3 and 4. Their squares are 12 = 1, 22 = 4, 32 = 9 and 42 = 16.
The required set = {12, 22, 32, 42} = { 1, 4, 9, 16 }
(iii) The even numbers between 1 and 9 are 2, 4, 6 and 8.
The required set = { 2, 4, 6, 8 }
(iv) The first eight letters of the English alphabet are a, b, c, d, e, f, g and h.
The required set = { a, b, c, d, e, f, g, h }
(v) The letters of the word 'BASKET' are B, A, S, K, E, T.
The required set = { b, a, s, k, e, t }
(vi) Four cities of India whose names start with the letter J are Jaipur, Jodhpur, Jalandhar and Jhansi.
The required set = { Jaipur, Jodhpur, Jalandhar, Jhansi }
(vii) Four closed geometrical figures are a triangle, a circle, a parallelogram and a pentagon. (Any four such figures may be taken.)
The required set = { triangle, circle, parallelogram, pentagon }
(viii) The vowels used in the word 'MONDAY' are O and A.
The required set = { o, a }
(ix) The single digit numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Out of these, the perfect squares are 0, 1, 4 and 9, since 0 = 02, 1 = 12, 4 = 22 and 9 = 32.
The required set = { 0, 1, 4, 9 }
Write each given set in the Set-Builder Form:
(i) { 2, 4, 6, 8, 10 }
(ii) { 2, 3, 5, 7, 11 }
(iii) { January, June, July }
(iv) { a, e, i, o, u }
(v) { Tuesday, Thursday }
(vi) { 1, 4, 9, 16, 25 }
(vii) { 5, 10, 15, 20, 25, 30 }
Answer
(i) The elements 2, 4, 6, 8 and 10 are the even natural numbers less than 12.
{ 2, 4, 6, 8, 10 } = { x : x is an even natural number less than 12 }
(ii) The elements 2, 3, 5, 7 and 11 are the prime numbers less than 12.
{ 2, 3, 5, 7, 11 } = { x : x is a prime number less than 12 }
(iii) The elements January, June and July are the months whose names start with the letter J.
{ January, June, July } = { x : x is a month whose name starts with the letter J }
(iv) The elements a, e, i, o and u are the vowels in the English alphabet.
{ a, e, i, o, u } = { x : x is a vowel in the English alphabet }
(v) The elements Tuesday and Thursday are the days of the week whose names start with the letter T.
{ Tuesday, Thursday } = { x : x is a day of the week whose name starts with the letter T }
(vi) The elements 1, 4, 9, 16 and 25 are the perfect square natural numbers up to 25.
{ 1, 4, 9, 16, 25 } = { x : x is a perfect square natural number up to 25 }
(vii) The elements 5, 10, 15, 20, 25 and 30 are the natural numbers up to 30 that are divisible by 5.
{ 5, 10, 15, 20, 25, 30 } = { x : x is a natural number up to 30 and divisible by 5 }
Write each of the following sets in Roster (tabular) Form and also in Set-Builder Form:
(i) Set of all natural numbers that can divide 24 completely.
(ii) Set of odd numbers between 20 and 35.
(iii) Set of letters used in the word 'CALCUTTA'.
(iv) Set of names of the first five months of a year.
(v) Set of all two-digit numbers that are perfect squares as well.
Answer
(i) The natural numbers that divide 24 completely are 1, 2, 3, 4, 6, 8, 12 and 24.
Roster form: { 1, 2, 3, 4, 6, 8, 12, 24 }
Set-builder form: { x : x is a natural number which divides 24 completely }
(ii) The odd numbers between 20 and 35 are 21, 23, 25, 27, 29, 31 and 33.
Roster form: { 21, 23, 25, 27, 29, 31, 33 }
Set-builder form: { x : x is an odd number between 20 and 35 }
(iii) The letters in the word 'CALCUTTA' are C, A, L, C, U, T, T, A. Writing each letter only once, we get C, A, L, U, T.
Roster form: { c, a, l, u, t }
Set-builder form: { x : x is a letter used in the word 'CALCUTTA' }
(iv) The first five months of a year are January, February, March, April and May.
Roster form: { January, February, March, April, May }
Set-builder form: { x : x is the name of first five months of a year }
(v) The two-digit numbers that are perfect squares are 42 = 16, 52 = 25, 62 = 36, 72 = 49, 82 = 64 and 92 = 81.
Roster form: { 16, 25, 36, 49, 64, 81 }
Set-builder form: { x : x is a perfect square two-digit number }
Write, in Roster Form, the set of:
(i) the first four odd natural numbers each divisible by 5.
(ii) the counting numbers between 15 and 35; each of which is divisible by 6.
(iii) the names of the last three days of a week.
(iv) the names of the last four months of a year.
Answer
(i) The first four odd natural numbers each divisible by 5 are 5, 15, 25 and 35.
The required set = { 5, 15, 25, 35 }
(ii) The counting numbers between 15 and 35 which are divisible by 6 are 18, 24 and 30.
The required set = { 18, 24, 30 }
(iii) The last three days of a week are Friday, Saturday and Sunday.
The required set = { Friday, Saturday, Sunday }
(iv) The last four months of a year are September, October, November and December.
The required set = { September, October, November, December }
State whether the given set is infinite or finite:
(i) { 3, 5, 7, ...... }
(ii) { 1, 2, 3, 4 }
(iii) { ..., -3, -2, -1, 0, 1, 2 }
(iv) { 20, 30, 40, 50, ..., 200 }
Answer
(i) The dots after 7 show that the elements continue without end.
Infinite set
(ii) The set has a definite (countable) number of elements, i.e. 4 elements.
Finite set
(iii) The dots before -3 show that the elements continue without end on the left.
Infinite set
(iv) The elements start at 20 and end at 200, so the number of elements is countable.
Finite set
Which of the following sets is empty?
(i) Set of counting numbers between 5 and 6.
(ii) Set of odd numbers between 7 and 19.
(iii) Set of odd numbers between 7 and 9.
(iv) Set of even numbers that are not divisible by 2.
(v) { 0 }
Answer
(i) There is no counting number between 5 and 6. So, this set is empty.
(ii) The odd numbers between 7 and 19 are 9, 11, 13, 15 and 17. So, this set is not empty.
(iii) There is no odd number between 7 and 9 (the only number between them is 8, which is even). So, this set is empty.
(iv) Every even number is divisible by 2, so there is no even number that is not divisible by 2. So, this set is empty.
(v) The set { 0 } has one element, namely 0. So, this set is not empty.
Hence, the sets in (i), (iii) and (iv) are empty.
State which pair of sets given below are equal sets and which are equivalent:
(i) { 3, 5, 7 } and { 5, 3, 7 }
(ii) { 8, 6, 10, 12 } and { 3, 2, 4, 6 }
(iii) { 7, 7, 2, 1, 2 } and { 1, 2, 7 }
(iv) { 2, 4, 6, 8, 10 } and { a, b, d, e, m }
Answer
(i) Both sets have exactly the same elements 3, 5 and 7.
Equal sets
(ii) Both sets have 4 elements each, but the elements are not the same.
Equivalent sets
(iii) { 7, 7, 2, 1, 2 } = { 1, 2, 7 } (repetition does not change a set), which has the same elements as { 1, 2, 7 }.
Equal sets
(iv) Both sets have 5 elements each, but the elements are not the same.
Equivalent sets
State which of the following are finite sets and which are infinite:
(i) Set of integers
(ii) { Multiples of 5 }
(iii) { Fractions between 1 and 2 }
(iv) { Number of people in India }
(v) Set of trees in the world
(vi) Set of leaves on a tree
(vii) Set of children in all the schools of Delhi
(viii) { ...., -4, -2, 0, 2, 4, 6, 8 }
(ix) { -12, -9, -6, -3, 0, 3, 6, ..... }
(x) { Number of points in a line segment 4 cm long }
Answer
(i) The integers are ......, -2, -1, 0, 1, 2, ......, which continue without end.
Infinite set
(ii) The multiples of 5 are 5, 10, 15, 20, ......, which continue without end.
Infinite set
(iii) Between any two numbers there are unlimited fractions.
Infinite set
(iv) The number of people in India is large but countable.
Finite set
(v) The number of trees in the world cannot be counted exactly.
Infinite set
(vi) The number of leaves on a tree is countable.
Finite set
(vii) The number of children in all the schools of Delhi is countable.
Finite set
(viii) The dots before -4 show that the elements continue without end on the left.
Infinite set
(ix) The dots after 6 show that the elements continue without end on the right.
Infinite set
(x) A line segment, however small, contains an unlimited number of points.
Infinite set
State whether or not the following sets are empty:
(i) { Prime numbers divisible by 2 }
(ii) { Negative natural numbers }
(iii) { Women with height 5 metres }
(iv) { Integers less than 5 }
(v) { Prime numbers between 17 and 23 }
(vi) Set of even numbers not divisible by 2
(vii) Set of multiples of 3 that are more than 9 and less than 15.
Answer
(i) 2 is a prime number which is divisible by 2. So, the set = { 2 }.
Not empty
(ii) The natural numbers are 1, 2, 3, ......; none of them is negative. So, there is no negative natural number.
Empty
(iii) There is no woman with a height of 5 metres.
Empty
(iv) The integers less than 5 are 4, 3, 2, 1, 0, -1, ......, which exist.
Not empty
(v) 19 is a prime number between 17 and 23. So, the set = { 19 }.
Not empty
(vi) Every even number is divisible by 2, so there is no even number that is not divisible by 2.
Empty
(vii) The only multiple of 3 that is more than 9 and less than 15 is 12. So, the set = { 12 }.
Not empty
State if the given pairs of sets are equal sets or equivalent sets:
(i) { Natural numbers less than five } and { Letters of the word 'BOAT' }.
(ii) { 2, 4, 6, 8, 10 } and { even natural numbers less than 12 }.
(iii) { 1, 3, 5, 7, ..... } and set of odd natural numbers.
(iv) { Letters of the word MEMBER } and { Letters of the word 'REMEMBER' }.
(v) { Negative natural numbers } and { 50th day of a month }
(vi) { Even natural numbers } and { Odd natural numbers }.
Answer
(i) { Natural numbers less than five } = { 1, 2, 3, 4 } and { Letters of the word 'BOAT' } = { B, O, A, T }. Both have 4 elements each, but the elements are not the same.
Equivalent sets
(ii) { 2, 4, 6, 8, 10 } and { even natural numbers less than 12 } = { 2, 4, 6, 8, 10 } have exactly the same elements.
Equal sets
(iii) { 1, 3, 5, 7, ..... } and the set of odd natural numbers = { 1, 3, 5, 7, ..... } have exactly the same elements.
Equal sets
(iv) { Letters of the word MEMBER } = { M, E, B, R } and { Letters of the word 'REMEMBER' } = { R, E, M, B } have exactly the same elements.
Equal sets
(v) { Negative natural numbers } = { } (there is no negative natural number) and { 50th day of a month } = { } (no month has a 50th day). Both are empty sets, and two empty sets are always equal.
Equal sets
(vi) { Even natural numbers } and { Odd natural numbers } are both infinite sets, so they have the same (infinite) number of elements, but the elements are not the same.
Equivalent sets
State whether the following are finite or infinite sets:
(i) { 2, 4, 6, 8, ...., 800 }
(ii) { ...., -5, -4, -3, -2 }
(iii) { x : x is an integer between -60 and 60 }
(iv) { No. of electrical appliances working in your house }
(v) { x : x is a whole number greater than 20 }
(vi) { x : x is a whole number less than 20 }
Answer
(i) The elements start at 2 and end at 800, so the number of elements is countable.
Finite set
(ii) The dots before -5 show that the elements continue without end on the left.
Infinite set
(iii) The integers between -60 and 60 are -59, -58, ......, 58, 59, which are countable.
Finite set
(iv) The number of electrical appliances working in your house is countable.
Finite set
(v) The whole numbers greater than 20 are 21, 22, 23, ......, which continue without end.
Infinite set
(vi) The whole numbers less than 20 are 0, 1, 2, ......, 19, which are countable.
Finite set
For each statement given below, write True or False:
(i) { ...., -8, -4, 0, 4, 8 } is a finite set.
(ii) { -32, -28, -24, -20, ...., 0, 4, 8, 16 } is an infinite set.
(iii) { x : x is a natural number less than 1 } is the empty set.
(iv) { Whole numbers between 15 and 16 } = { Natural numbers between 5 and 6 }.
(v) { Odd numbers divisible by 2 } is the empty set.
(vi) { Even natural numbers divisible by 3 } is the empty set.
(vii) { x : x is positive and x < 0 } is the empty set.
(viii) { ...., -5, -3, -1, 1, 3, 5, .... } is a finite set.
Answer
(i) The dots before -8 show that the elements continue without end, so the set is infinite, not finite.
False
(ii) The elements start at -32 and end at 16, so the number of elements is countable. So, it is a finite set, not an infinite set.
False
(iii) There is no natural number less than 1, so the set has no element.
True
(iv) There is no whole number between 15 and 16, and there is no natural number between 5 and 6. So, both sets are the empty set and are therefore equal.
True
(v) No odd number is divisible by 2, so the set has no element.
True
(vi) 6, 12, 18, 24, ...... are even natural numbers divisible by 3, so the set is not empty.
False
(vii) No positive number can be less than 0, so the set has no element.
True
(viii) The dots on both sides show that the elements continue without end, so the set is infinite, not finite.
False
State, giving reasons, which of the following pairs of sets are disjoint sets and which are overlapping sets:
(i) A = { Girls with ages below 15 years } and
B = { Girls with ages above 15 years }
(ii) C = { Boys with ages above 20 years } and
D = { Boys with ages above 27 years }
(iii) A = { Natural numbers between 35 and 60 } and
B = { Natural numbers between 50 and 80 }
(iv) P = { Students of Class IX studying in I.C.S.E. Board } and
Q = { Students of Class IX }
(v) A = { Natural numbers that are multiples of 3 and less than 30 } and
B = { Natural numbers divisible by 4 and lying between 20 and 45 }
(vi) P = { Letters in the word 'ALLAHABAD' } and
Q = { Letters in the word 'NAINITAL' }
Answer
Overlapping sets : Two sets are overlapping if they have at least one element in common.
Disjoint sets : Two sets are disjoint if they have no element in common.
(i) Disjoint sets; as no girl can be of age below 15 years and also above 15 years, the two sets have no element in common.
(ii) Overlapping sets; as boys with ages above 27 years are also above 20 years, so they are common to both the sets.
(iii) Overlapping sets; as the natural numbers from 51 to 59 are common to both the sets.
(iv) Overlapping sets; as the students of Class IX studying in I.C.S.E. Board are common to both the sets.
(v) Overlapping sets; as A = { 3, 6, 9, ......, 27 } and B = { 24, 28, 32, 36, 40, 44 }. The number 24 is common to both the sets.
(vi) Overlapping sets; as P = { A, L, H, B, D } and Q = { N, A, I, T, L }. The letters A and L are common to both the sets.
Write the cardinal number of each of the following sets:
(i) A = { 0, 1, 2, 4 }
(ii) B = { -3, -1, 1, 3, 5, 7 }
(iii) C = { }
(iv) D = { 3, 2, 2, 1, 3, 1, 2 }
(v) E = { Natural numbers between 15 and 20 }
(vi) F = { Whole numbers from 8 to 14 }
Answer
(i) A = { 0, 1, 2, 4 } has 4 elements.
Hence, n(A) = 4.
(ii) B = { -3, -1, 1, 3, 5, 7 } has 6 elements.
Hence, n(B) = 6.
(iii) C = { } is the empty set, which has no element.
Hence, n(C) = 0.
(iv) D = { 3, 2, 2, 1, 3, 1, 2 } = { 1, 2, 3 } has 3 elements.
Hence, n(D) = 3.
(v) The natural numbers between 15 and 20 are 16, 17, 18 and 19. So, E = { 16, 17, 18, 19 }, which has 4 elements.
Hence, n(E) = 4.
(vi) The whole numbers from 8 to 14 are 8, 9, 10, 11, 12, 13 and 14. So, F = { 8, 9, 10, 11, 12, 13, 14 }, which has 7 elements.
Hence, n(F) = 7.
Given:
A = { Natural numbers less than 10 }
B = { Letters of the word 'INVENTION' }
C = { Squares of the first four whole numbers }
D = { Odd numbers divisible by 2 }
Find:
(i) n(A)
(ii) n(B)
(iii) n(C)
(iv) n(D)
Answer
(i) The natural numbers less than 10 are 1, 2, 3, 4, 5, 6, 7, 8 and 9. So, A = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }, which has 9 elements.
Hence, n(A) = 9.
(ii) The letters of the word 'INVENTION' are I, N, V, E, N, T, I, O, N. Writing each letter only once, we get I, N, V, E, T, O. So, B = { I, N, V, E, T, O }, which has 6 elements.
Hence, n(B) = 6.
(iii) The first four whole numbers are 0, 1, 2 and 3. Their squares are 02 = 0, 12 = 1, 22 = 4 and 32 = 9. So, C = { 0, 1, 4, 9 }, which has 4 elements.
Hence, n(C) = 4.
(iv) No odd number is divisible by 2, so D = { }, which has no element.
Hence, n(D) = 0.
State true or false for each of the following. Correct the wrong statement.
(i) If A = { 0 }, then n(A) = 0.
(ii) n(∅) = 1.
(iii) If T = { a, l, a, h, b, d, h }; then n(T) = 5
(iv) If B = { 1, 5, 51, 15, 5, 1 }, then n(B) = 6.
Answer
(i) A = { 0 } has one element, namely 0. So, n(A) = 1, not 0.
False; the correct statement is n(A) = 1.
(ii) The empty set has no element. So, n(∅) = 0, not 1.
False; the correct statement is n(∅) = 0.
(iii) T = { a, l, a, h, b, d, h } = { a, l, h, b, d }, which has 5 elements. So, n(T) = 5.
True
(iv) B = { 1, 5, 51, 15, 5, 1 } = { 1, 5, 51, 15 }, which has 4 elements. So, n(B) = 4, not 6.
False; the correct statement is n(B) = 4.
A collection of beautiful flowers is a set:
Yes
No
none of these two
Answer
The word 'beautiful' is a relative term. A flower that is beautiful to one person may not be beautiful to another person. So, the objects are not well defined and the collection does not form a set.
Hence, option 2 is the correct option.
Set of integers between 5 and 6 is:
an infinite set
not an empty set
a singleton set
the empty set
Answer
There is no integer between 5 and 6. So, the set has no element, i.e. it is the empty set.
Hence, option 4 is the correct option.
Set A = Set B ⇒
A and B are equivalent sets
A and B have identical elements
A contains all the elements of B and B contains all the elements of A.
all of the above are true
Answer
If Set A = Set B, then both sets have exactly the same (identical) elements. This means A contains all the elements of B and B contains all the elements of A, and since the number of elements is also the same, A and B are equivalent too. So, all the given statements are true.
Hence, option 4 is the correct option.
n(A) = n(B) ⇒
A = B
A ≠ B
A contains all the elements of B and B contains all the elements of A.
none of these
Answer
n(A) = n(B) means A and B have the same number of elements, i.e. they are equivalent sets. The elements need not be the same, so A and B need not be equal, need not be unequal, and need not contain all the elements of each other. So, none of the given statements is necessarily true.
Hence, option 4 is the correct option.
Number of elements in { prime numbers between the integers 7 and 11 } is:
0
3
5
2
Answer
The integers between 7 and 11 are 8, 9 and 10. None of these is a prime number. So, the set has no element.
Hence, option 1 is the correct option.
If A = { 0, 1, 2, { }, {0} }, then n(A) is:
5
4
3
6
Answer
The elements of A are 0, 1, 2, { } and {0}, which are 5 distinct elements.
Hence, option 1 is the correct option.
N = { x : x is a natural number and x2 ≤ 5 }, then set N is:
{-2, -1, 0, 1, 2}
{0, 1, 2}
{1, 2}
{-2, -1, 0}
Answer
The natural numbers are 1, 2, 3, ......
For x = 1, x2 = 1 ≤ 5
For x = 2, x2 = 4 ≤ 5
For x = 3, x2 = 9 > 5
So, N = { 1, 2 }.
Hence, option 3 is the correct option.
W = { x is a whole number and x2 ≤ 5 }, then set W is equal to:
{-2, -1, 0, 1, 2}
{0, 1, 2}
{1, 2}
{-2, -1, 0}
Answer
The whole numbers are 0, 1, 2, 3, ......
For x = 0, x2 = 0 ≤ 5
For x = 1, x2 = 1 ≤ 5
For x = 2, x2 = 4 ≤ 5
For x = 3, x2 = 9 > 5
So, W = { 0, 1, 2 }.
Hence, option 2 is the correct option.
I = { x is an integer and x2 ≤ 5 }, then set I is equal to:
{-2, -1, 0, 1, 2}
{0, 1, 2}
{1, 2}
{-2, -1, 0}
Answer
The integers are ......, -2, -1, 0, 1, 2, ......
For x = 0, x2 = 0 ≤ 5
For x = ±1, x2 = 1 ≤ 5
For x = ±2, x2 = 4 ≤ 5
For x = ±3, x2 = 9 > 5
So, I = { -2, -1, 0, 1, 2 }.
Hence, option 1 is the correct option.
If A = { 1, 2, 3, 4, 5, ......... } and B = { 2, 4, 6, 10, 12 }, then sets A and B are:
disjoint
overlapping
finite
equivalent
Answer
A = { 1, 2, 3, 4, 5, ......... } is the set of natural numbers and B = { 2, 4, 6, 10, 12 }. The elements 2, 4, 6, 10 and 12 of B are all present in A, so the two sets have elements in common.
Hence, option 2 is the correct option.
Statement 1: Sets { } and {0} are equal sets.
Statement 2: { } ≠ {0}.
Which of the following options is correct?
Both the statements are true.
Both the statements are false.
Statement 1 is true, and statement 2 is false.
Statement 1 is false, and statement 2 is true.
Answer
Statement 1: { } is the empty set and contains no element, whereas {0} contains the element 0. So, { } and {0} are not equal. Hence, Statement 1 is false.
Statement 2: Since { } has no element and {0} has the element 0, the two sets are not equal, i.e. { } ≠ {0}. Hence, Statement 2 is true.
Hence, option 4 is the correct option.
Statement 1: Two sets A and B are overlapping if they have atleast one element in common.
Statement 2: If set A = { even natural numbers } and set B = { odd natural numbers }, sets A and B are disjoint.
Which of the following options is correct?
Both the statements are true.
Both the statements are false.
Statement 1 is true, and statement 2 is false.
Statement 1 is false, and statement 2 is true.
Answer
Statement 1: By definition, two sets are overlapping if they have at least one element in common. Hence, Statement 1 is true.
Statement 2: A = { even natural numbers } and B = { odd natural numbers } have no element in common, so they are disjoint sets. Hence, Statement 2 is true.
Hence, option 1 is the correct option.
Assertion (A): {2, 4, 6, ....} and {1, 3, 5, ....} are equivalent sets.
Reason (R): {2, 4, 6, .....} has an infinite number of elements and {1, 3, 5, ......} also has an infinite number of elements.
A is true, R is false.
A is false, R is true.
Both A and R are true.
Both A and R are false.
Answer
Assertion (A): {2, 4, 6, ....} is the set of even natural numbers and {1, 3, 5, ....} is the set of odd natural numbers. Both are infinite sets, and two infinite sets are always equivalent. So, Assertion (A) is true.
Reason (R): Both {2, 4, 6, .....} and {1, 3, 5, ......} have an infinite number of elements. So, Reason (R) is true, and it correctly explains the Assertion.
Hence, option 3 is the correct option.
Assertion (A): Sets A and B are equal if all elements of A are in B and all elements of B are in A.
Reason (R): If two sets have identical elements, they are equal.
A is true, R is false.
A is false, R is true.
Both A and R are true.
Both A and R are false.
Answer
Assertion (A): Two sets are equal if all elements of A are in B and all elements of B are in A, i.e. the two sets have identical elements. So, Assertion (A) is true.
Reason (R): If two sets have identical elements, they are equal. This is the definition of equal sets, so Reason (R) is true, and it correctly explains the Assertion.
Hence, option 3 is the correct option.