Find, whether or not, each of the following collections represent a set :
(i) The collection of good students in your school.
(ii) The collection of the numbers between 30 and 45.
(iii) The collection of tall people in your colony.
(iv) The collection of interesting books in your school library.
(v) The collection of books in the library and are of your interest.
Answer
A collection is a set only if it is a well defined collection of objects.
(i) The collection of good students in your school — The word 'good' is not well defined as it varies from person to person. It is not a set.
(ii) The collection of the numbers between 30 and 45 — This is a well defined collection. It is a set.
(iii) The collection of tall people in your colony — The word 'tall' is not well defined as it varies from person to person. It is not a set.
(iv) The collection of interesting books in your school library — The word 'interesting' is not well defined as a book may be interesting to one person and not to another. It is not a set.
(v) The collection of books in the library and are of your interest — Here the books are described with reference to your own interest, so for any given book it can be decided, without doubt, whether it belongs to the collection or not. This is well defined. It is a set.
Hence, (ii) and (v) are sets; whereas (i), (iii) and (iv) are not sets because these collections are not well defined.
State whether true or false :
(i) Set {4, 5, 8} is same as the set {5, 4, 8} and the set {8, 4, 5}.
(ii) Sets {a, b, m, n} and {a, a, m, b, n, n} are same.
(iii) Set of letters in the word 'suchismita' is {s, u, c, h, i, m, t, a}.
(iv) Set of letters in the word 'MAHMOOD' is {M, A, H, O, D}.
Answer
(i) Set {4, 5, 8} is same as the set {5, 4, 8} and the set {8, 4, 5}.
The elements of a set can be written in any order, so all three sets have the same elements.
True
(ii) Sets {a, b, m, n} and {a, a, m, b, n, n} are same.
Repeated elements are written only once, so {a, a, m, b, n, n} = {a, b, m, n}.
True
(iii) Set of letters in the word 'suchismita' is {s, u, c, h, i, m, t, a}.
The letters of 'suchismita' are s, u, c, h, i, s, m, i, t, a. Writing each letter only once, we get {s, u, c, h, i, m, t, a}.
True
(iv) Set of letters in the word 'MAHMOOD' is {M, A, H, O, D}.
The letters of 'MAHMOOD' are M, A, H, M, O, O, D. Writing each letter only once, we get {M, A, H, O, D}.
True
Let set A = {6, 8, 10, 12} and set B = {3, 9, 15, 18}.
Insert the symbol '∈' or '∉' to make each of the following true :
(i) 6 .... A
(ii) 10 .... B
(iii) 18 .... B
(iv) (6 + 3) .... B
(v) (15 − 9) .... B
(vi) 12 .... A
(vii) (6 + 8) .... A
(viii) 6 and 8 .... A
Answer
(i) Since 6 is an element of A, 6 ∈ A
(ii) Since 10 is not an element of B, 10 ∉ B
(iii) Since 18 is an element of B, 18 ∈ B
(iv) 6 + 3 = 9, which is an element of B, (6 + 3) ∈ B
(v) 15 − 9 = 6, which is not an element of B, (15 − 9) ∉ B
(vi) Since 12 is an element of A, 12 ∈ A
(vii) 6 + 8 = 14, which is not an element of A, (6 + 8) ∉ A
(viii) Since both 6 and 8 are elements of A, 6 and 8 ∈ A
Express each of the following sets in roster form :
(i) Set of odd whole numbers between 15 and 27.
(ii) A = Set of letters in the word "CHITAMBARAM"
(iii) B = {All even numbers from 15 to 26}
(iv) P = {x : x is a vowel used in the word 'ARITHMETIC'}
(v) S = {Squares of the first eight whole numbers}
(vi) Set of all integers between 7 and 94 which are divisible by 6.
(vii) C = {All composite numbers between 2 and 20}
(viii) D = Set of prime numbers from 2 to 23.
(ix) E = Set of natural numbers below 30 which are divisible by 2 or 5.
(x) F = Set of factors of 24.
(xi) G = Set of names of three closed figures in Geometry.
(xii) H = {x : x ∈ W and x < 10}
(xiii) J = {x : x ∈ N and 2x − 3 ≤ 17}
(xiv) K = {x : x is an integer and − 3 < x < 5}
Answer
(i) The odd whole numbers between 15 and 27 are 17, 19, 21, 23 and 25.
{17, 19, 21, 23, 25}
(ii) The letters of the word CHITAMBARAM are C, H, I, T, A, M, B, A, R, A, M. Writing each letter only once, we get C, H, I, T, A, M, B and R.
A = {c, h, i, t, a, m, b, r}
(iii) The even numbers from 15 to 26 are 16, 18, 20, 22, 24 and 26.
B = {16, 18, 20, 22, 24, 26}
(iv) The vowels used in the word ARITHMETIC are A, I, E, I. Writing each vowel only once, we get A, I and E.
P = {a, i, e}
(v) The first eight whole numbers are 0, 1, 2, 3, 4, 5, 6 and 7. Their squares are 0, 1, 4, 9, 16, 25, 36 and 49.
S = {0, 1, 4, 9, 16, 25, 36, 49}
(vi) The integers between 7 and 94 which are divisible by 6 are 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84 and 90.
{12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90}
(vii) The composite numbers between 2 and 20 are 4, 6, 8, 9, 10, 12, 14, 15, 16 and 18.
C = {4, 6, 8, 9, 10, 12, 14, 15, 16, 18}
(viii) The prime numbers from 2 to 23 are 2, 3, 5, 7, 11, 13, 17, 19 and 23.
D = {2, 3, 5, 7, 11, 13, 17, 19, 23}
(ix) The natural numbers below 30 which are divisible by 2 are 2, 4, 6, 8, ..., 28 and those divisible by 5 are 5, 10, 15, 20, 25. Taking all such numbers, we get 2, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 25, 26 and 28.
E = {2, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 25, 26, 28}
(x) The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
F = {1, 2, 3, 4, 6, 8, 12, 24}
(xi) Three closed figures in Geometry are a triangle, a circle and a square.
G = {triangle, circle, square}
(xii) Here x ∈ W and x < 10, so x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
H = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
(xiii) Here x ∈ N and 2x − 3 ≤ 17
As, 2x − 3 ≤ 17
⇒ 2x ≤ 17 + 3
⇒ 2x ≤ 20
⇒ x ≤ 10
So x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
J = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(xiv) Here x is an integer and − 3 < x < 5, so x = −2, −1, 0, 1, 2, 3, 4
K = {−2, −1, 0, 1, 2, 3, 4}
Express each of the following sets in set-builder notation (form) :
(i) {3, 6, 9, 12, 15}
(ii) {2, 3, 5, 7, 11, 13, ...}
(iii) {1, 4, 9, 16, 25, 36}
(iv) {0, 2, 4, 6, 8, 10, 12, ...}
(v) {Monday, Tuesday, Wednesday}
(vi) {23, 25, 27, 29, ...}
(vii)
(viii) {42, 49, 56, 63, 70, 77}
Answer
(i) {3, 6, 9, 12, 15}
These are natural numbers divisible by 3 which are less than 18.
{x : x is a natural number divisible by 3; x < 18}
(ii) {2, 3, 5, 7, 11, 13, ...}
These are the prime numbers.
{x : x is a prime number}
(iii) {1, 4, 9, 16, 25, 36}
These are perfect squares of natural numbers up to 6, i.e. 12, 22, 32, 42, 52, 62.
{x : x is a perfect square natural number and x ≤ 36}
(iv) {0, 2, 4, 6, 8, 10, 12, ...}
These are whole numbers divisible by 2.
{x : x is a whole number divisible by 2}
(v) {Monday, Tuesday, Wednesday}
These are the first three days of the week.
{x : x is one of the first three days of the week}
(vi) {23, 25, 27, 29, ...}
These are odd natural numbers greater than or equal to 23.
{x : x is an odd natural number; x ≥ 23}
(vii)
Each element is of the form where n takes the natural number values from 3 to 8.
{x : x = , where n is a natural number; 3 ≤ n ≤ 8}
(viii) {42, 49, 56, 63, 70, 77}
These are natural numbers divisible by 7 lying between 42 to 77 inclusive.
{x : x is a natural number divisible by 7; 42 ≤ x ≤ 77}
Given : A = {x : x is a multiple of 2 and is less than 25}
B = {x : x is a square of a natural number and is less than 25}
C = {x : x is a multiple of 3 and is less than 25}
D = {x : x is a prime number less than 25}
Write the sets A, B, C and D in roster form.
Answer
A = {x : x is a multiple of 2 and is less than 25}
The multiples of 2 less than 25 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 and 24.
A = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24}
B = {x : x is a square of a natural number and is less than 25}
The squares of natural numbers less than 25 are 12 = 1, 22 = 4, 32 = 9 and 42 = 16 (since 52 = 25 is not less than 25).
B = {1, 4, 9, 16}
C = {x : x is a multiple of 3 and is less than 25}
The multiples of 3 less than 25 are 3, 6, 9, 12, 15, 18, 21 and 24.
C = {3, 6, 9, 12, 15, 18, 21, 24}
D = {x : x is a prime number less than 25}
The prime numbers less than 25 are 2, 3, 5, 7, 11, 13, 17, 19 and 23.
D = {2, 3, 5, 7, 11, 13, 17, 19, 23}