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, 9

This 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}

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