RD Chapter 1 Sets Ex 1.2 Solutions
Question - 1 : - Describe the following sets in Roster form:
(i) {x : x is a letter before e in the English alphabet}
(ii) {x ∈ N: x2 < 25}
(iii) {x ∈ N: x is a prime number, 10 < x < 20}
(iv) {x ∈ N: x = 2n, n ∈ N}
(v) {x ∈ R: x > x}
(vi) {x : x is a prime number which is a divisor of 60}
(vii) {x : x is a two digit number such that the sum of its digits is 8}
(viii) The set of all letters in the word ‘Trigonometry’
(ix) The set of all letters in the word ‘Better.’
Answer - 1 : -
(i) {x: x is a letter before e in the English alphabet}
So, when we read whole sentence it becomes x is such that x is a letterbefore ‘e’ in the English alphabet. Now letters before ‘e’ are a,b,c,d.
∴ Rosterform will be {a,b,c,d}.
(ii) {x ∈N: x2 < 25}
x ∈ N that implies x is a natural number.
x2 < 25
x < ±5
As x belongs to the natural number that means x < 5.
All numbers less than 5 are 1,2,3,4.
∴ Rosterform will be {1,2,3,4}.
(iii) {x∈ N:x is a prime number, 10 < x < 20}
X is a natural number and is between 10 and 20.
X is such that X is a prime number between 10 and 20.
Prime numbers between 10 and 20 are 11,13,17,19.
∴ Rosterform will be {11,13,17,19}.
(iv) {x∈ N:x = 2n, n ∈ N}
X is a natural number also x = 2n
∴ Rosterform will be {2,4,6,8…..}.
This an infinite set.
(v) {x∈ R:x > x}
Any real number is equal to its value it is neither less nor greater.
So, Roster form of such real numbers which has value less than itself hasno such numbers.
∴ Rosterform will be ϕ. This is called a null set.
(vi) {x: x is a prime number which is a divisor of 60}
All numbers which are divisor of 60 are = 1,2,3,4,5,6,10,12,15,20,30,60.
Now, prime numbers are = 2, 3, 5.
∴ Rosterform will be {2, 3, 5}.
(vii) {x: x is a two digit number such that the sum of its digits is 8}
Numbers which have sum of its digits as 8 are = 17, 26, 35, 44, 53, 62,71, 80
∴Roster form will be {17, 26, 35, 44, 53, 62, 71, 80}.
(viii)The set of all letters in the word ‘Trigonometry’
As repetition is not allowed in a set, then the distinct letters are
Trigonometry = t, r, i, g, o, n, m, e, y
∴ Rosterform will be {t, r, i, g, o, n, m, e, y}
(ix)The set of all letters in the word ‘Better.’
As repetition is not allowed in a set, then the distinct letters are
Better = b, e, t, r
∴ Rosterform will be {b, e, t, r}
Question - 2 : - Describe the following sets in set-builder form:
(i) A = {1, 2, 3, 4, 5, 6}
(ii) B = {1, 1/2, 1/3, 1/4, 1/5, …..}
(iii) C = {0, 3, 6, 9, 12,….}
(iv) D = {10, 11, 12, 13, 14, 15}
(v) E = {0}
(vi) {1, 4, 9, 16,…,100}
(vii) {2, 4, 6, 8,….}
(viii) {5, 25, 125, 625}
Answer - 2 : -
(i) A= {1, 2, 3, 4, 5, 6}
{x : x ∈ N, x<7}
This is read as x is such that x belongs to natural number and x is lessthan 7. It satisfies all condition of roster form.
(ii) B= {1, 1/2, 1/3, 1/4, 1/5, …}
{x : x = 1/n, n ∈ N}
This is read as x is such that x =1/n, where n ∈ N.
(iii) C= {0, 3, 6, 9, 12, ….}
{x : x = 3n, n ∈ Z+,the set of positive integers}
This is read as x is such that C is the set of multiples of 3 including 0.
(iv) D= {10, 11, 12, 13, 14, 15}
{x : x ∈ N, 9This is read as x is such that D is the set of natural numbers which aremore than 9 but less than 16.
(v) E= {0}
{x : x = 0}
This is read as x is such that E is an integer equal to 0.
(vi) {1,4, 9, 16,…, 100}
Where,
12 = 1
22 = 4
32 = 9
42 = 16
.
.
.
102 = 100
So, above set can be expressed in set-builder form as {x2: x ∈ N, 1≤ x ≤10}
(vii) {2,4, 6, 8,….}
{x: x = 2n, n ∈ N}
This is read as x is such that the given set are multiples of 2.
(viii) {5,25, 125, 625}
Where,
51 = 5
52 = 25
53 = 125
54 = 625
So, above set can be expressed in set-builder form as {5n: n ∈ N, 1≤ n ≤ 4}
Question - 3 : - List all the elements of the following sets:
(i) A={x : x2≤ 10, x ∈ Z}
(ii) B = {x : x = 1/(2n-1), 1 ≤ n ≤ 5}
(iii) C = {x : x is an integer, -1/2 < x < 9/2}
(iv) D={x : x is a vowel in the word “EQUATION”}
(v) E = {x : x is a month of a year not having 31 days}
(vi) F={x : x is a letter of the word “MISSISSIPPI”}
Answer - 3 : -
(i) A={x: x2≤ 10, x ∈Z}
First of all, x is an integer hence it can be positive and negative also.
x2 ≤ 10
(-3)2 = 9 < 10
(-2)2 = 4 < 10
(-1)2 = 1 < 10
02 = 0 < 10
12 = 1 < 10
22 = 4 < 10
32 = 9 < 10
Square root of next integers are greater than 10.
x ≤ √10
x = 0, ±1, ±2, ±3
A = {0, ±1, ±2, ±3}
(ii) B= {x : x = 1/(2n-1), 1 ≤ n ≤ 5}
Let us substitute the value of n to find the values of x.
At n=1, x = 1/(2(1)-1) = 1/1
At n=2, x = 1/(2(2)-1) = 1/3
At n=3, x = 1/(2(3)-1) = 1/5
At n=4, x = 1/(2(4)-1) = 1/7
At n=5, x = 1/(2(5)-1) = 1/9
x = 1, 1/3, 1/5, 1/7, 1/9
∴B = {1, 1/3, 1/5, 1/7, 1/9}
(iii) C= {x : x is an integer, -1/2 < x < 9/2}
Given, x is an integer between -1/2 and 9/2
So all integers between -0.5∴ C= {0, 1, 2, 3, 4}
(iv) D={x: x is a vowel in the word “EQUATION”}
All vowels in the word ‘EQUATION’ are E, U, A, I, O
∴ D= {A, E, I, O, U}
(v) E= {x : x is a month of a year not having 31 days}
A month has either 28, 29, 30, 31 days.
Out of 12 months in a year which are not having 31 days are:
February, April, June, September, November.
∴E: {February, April, June, September, November}
(vi) F= {x : x is a letter of the word “MISSISSIPPI”}
Letters in word ‘MISSISSIPPI’ are M, I, S, P.
∴F = {M, I, S, P}.
Question - 4 : - Match each of the sets on the left in the roster form with the same set on the right described in the set-builder form:
(i) {A,P,L,E} (i) {x : x+5=5, x ∈ z}
(ii) {5,-5} (ii) {x : x is a prime natural number and a divisor of 10}
(iii) {0} (iii) {x : x is a letter of the word “RAJASTHAN”}
(iv) {1, 2, 5, 10} (iv) {x : x is a natural and divisor of 10}
(v) {A, H, J, R, S, T, N} (v) {x : x2 – 25 =0}
(vi) {2,5} (vi) {x : x is a letter of word “APPLE”}
Answer - 4 : -
(i) {A,P, L, E} ⇔ {x:x is a letter of word “APPLE”}
(ii) {5,-5} ⇔ {x:x2 – 25 =0}
The solution set of x2 –25 = 0 is x = ±5
(iii) {0} ⇔ {x:x+5=5, x ∈ z}
The solution set of x + 5 = 5 is x = 0.
(iv) {1,2, 5, 10} ⇔ {x: x is a natural and divisor of 10}
The natural numbers which are divisor of 10 are 1, 2, 5, 10.
(v) {A,H, J, R, S, T, N} ⇔ {x: x is a letter of the word “RAJASTHAN”}
The distinct letters of word “RAJASTHAN” are A, H, J, R, S, T, N.
(vi) {2,5} ⇔ {x:x is a prime natural number and a divisor of 10}
The prime natural numbers which are divisor of 10 are 2, 5.
Question - 5 : - Write the set of all vowels in the English alphabet which precede q.
Answer - 5 : -
Set of all vowels which precede q are
A, E, I, O these are the vowels they come before q.
∴ B = {A, E, I, O}.
Question - 6 : - Write the set of all positive integers whose cube is odd.
Answer - 6 : -
Every odd number has an odd cube
Odd numbers can be represented as 2n+1.
{2n+1: n ∈ Z, n>0} or
{1,3,5,7,……}
Question - 7 : - Write the set {1/2, 2/5, 3/10, 4/17, 5/26, 6/37, 7/50} in the set-builder form.
Answer - 7 : -
Where,
2 = 12 + 1
5 = 22 + 1
10 = 32 + 1
.
.
50 = 72 + 1
Here we can see denominator is square of numerator +1.
So, we can write the set builder form as
{n/(n2+1): n ∈ N, 1≤ n≤ 7}