Sets - Exercise 6(A)

Which of the following collections are sets ?

(i) All books in your school library.

(ii) All red flowers in a park.

(iii) All good players in your school.

(iv) All fiction movies.

(v) All easy problems in your book on mathematics.

(vi) All poor people in Mumbai.

(vii) All boys in your class weighing less than 50 kg.

(viii) All persons of repute in your colony.

(ix) All even numbers greater than 100.

(x) All integers less than -5.

Answer

(i) All books in your school library.

The collection is well-defined because one can objectively check if a book belongs to the library's catalog.

∴ It is a set.

(ii) All red flowers in a park.

Colour is a physical property that can be objectively identified.

∴ It is a set.

(iii) All good players in your school.

The term "good" is subjective; a player considered good by one person might not be considered good by another.

∴ It is not a set.

(iv) All fiction movies.

The genre of a movie is a defined category.

∴ It is a set.

(v) All easy problems in your book on mathematics.

The term "easy" is relative; a problem that is easy for one student may be difficult for another.

∴ It is not a set.

(vi) All poor people in Mumbai.

"Poor" is a relative term unless a specific economic threshold (like a poverty line) is provided.

∴ It is not a set.

(vii) All boys in your class weighing less than 50 kg.

Weight is a measurable, objective quantity.

∴ It is a set.

(viii) All persons of repute in your colony.

"Repute" (reputation) is based on personal opinion and varies from person to person.

∴ It is not a set.

(ix) All even numbers greater than 100.

The criteria for being even and being greater than 100 are mathematically definite.

∴ It is a set.

(x) All integers less than -5.

Integers less than -5 (such as -6, -7, .....) are clearly defined.

∴ It is a set.

Write each of the following sets in Roster form :

(i) A = set of all prime numbers between 70 and 100.

(ii) B = set of all whole numbers less than 8.

(iii) C = set of all integers lying between -7 and 2.

(iv) D = set of all composite numbers between 23 and 33.

(v) E = set of letters in the word, 'MATHEMATICS'.

(vi) F = set of consonants in the word, 'SECONDARY'.

(vii) G = set of vowels in the word, 'INTERMEDIATE'.

Answer

(i) A = set of all prime numbers between 70 and 100.

Prime numbers are numbers that have only two factors: 1 and the number itself.

∴ A = {71, 73, 79, 83, 89, 97}

(ii) B = set of all whole numbers less than 8.

Whole numbers start from 0.

∴ B = {0, 1, 2, 3, 4, 5, 6, 7}

(iii) C = set of all integers lying between -7 and 2.

"Between" means we do not include -7 or 2.

∴ C = {-6, -5, -4, -3, -2, -1, 0, 1}

(iv) D = set of all composite numbers between 23 and 33.

Composite numbers have more than two factors.

∴ D = {24, 25, 26, 27, 28, 30, 32}

(v) E = set of letters in the word, 'MATHEMATICS'.

We list each unique letter once, even if it appears multiple times in the word.

∴ E = {M, A, T, H, E, I, C, S}

(vi) F = set of consonants in the word, 'SECONDARY'.

Consonants are letters that are not vowels (a, e, i, o, u).

∴ F = {S, C, N, D, R, Y}

(vii) G = set of vowels in the word, 'INTERMEDIATE'.

Vowels are a, e, i, o, u. We list each unique vowel once.

∴ G = {I, E, A}

Write each of the following sets in Roster form and write the cardinal number of each :

(i) A = {x : x is an integer, -3 < x ≤ 4}.

(ii) B = {x : x ∈ N, 3x - 6 < 9}.

(iii) C = {x : x = n², n ∈ N, 10 < n < 16}.

(iv) D = {x : x ∈ W, x - 3 < 2}.

(v) E = {x : x = 2n - 1, n ∈ N and n < 6}.

(vi) F = {x : x is a letter in the word 'COMMON'}.

(vii) G = {x : x is a primary colour}.

(viii) H = {x : x is a digit in the numeral 2362}.

(ix) J = $\Big\lbrace x : x = \dfrac{1}{n}, n ∈ N, 4 \lt n \lt 10\Big\rbrace$.

Answer

(i) A = {x : x is an integer, -3 < x ≤ 4}.

Integers greater than -3 and less than or equal to 4 are -2, -1, 0, 1, 2, 3, 4.

A = {-2, -1, 0, 1, 2, 3, 4}, n(A) = 7

(ii) B = {x : x ∈ N, 3x - 6 < 9}.

3x - 6 < 9

⇒ 3x < 9 + 6

⇒ 3x < 15

⇒ x < $\dfrac{15}{3}$

⇒ x < 5.

Since x is a natural number (N), x can be 1, 2, 3, 4.

B = {1, 2, 3, 4}, n(B) = 4

(iii) C = {x : x = n², n ∈ N, 10 < n < 16}.

n can be 11, 12, 13, 14, 15.

Calculating x(n²) = (11)², (12)², (13)², (14)², (15)²

= (11 x 11), (12 x 12), (13 x 13), (14 x 14), (15 x 15)

= 121, 144, 169, 196, 225.

C = {121, 144, 169, 196, 225}, n(C) = 5

(iv) D = {x : x ∈ W, x - 3 < 2}.

x - 3 < 2

⇒ x < 2 + 3

⇒ x < 5

Since x is a whole number (W), x can be 0, 1, 2, 3, 4.

D = {0, 1, 2, 3, 4}, n(D) = 5

(v) E = {x : x = 2n - 1, n ∈ N and n < 6}.

n < 6, so n = 1, 2, 3, 4, 5.

Calculating x = (2n - 1):

For n = 1, x = 2(1) - 1 = 1

For n = 2, x = 2(2) - 1 = 3

For n = 3, x = 2(3) - 1 = 5

For n = 4, x = 2(4) - 1 = 7

For n = 5, x = 2(5) - 1 = 9

E = {1, 3, 5, 7, 9}, n(E) = 5

(vi) F = {x : x is a letter in the word 'COMMON'}.

The letters in 'COMMON' are C, O, M, M, O, N.

Removing repeated letters, we get C, O, M, N.

F = {C, O, M, N}, n(F) = 4

(vii) G = {x : x is a primary colour}.

The primary colours are Red, Blue, and Yellow.

G = {Red, Blue, Yellow}, n(G) = 3

(viii) H = {x : x is a digit in the numeral 2362}.

The digits in 2362 are 2, 3, 6, 2.

Removing the repeated digit '2', we get 2, 3, 6.

H = {2, 3, 6}, n(H) = 3

(ix) J = $\Big\lbrace x : x = \dfrac{1}{n}, n ∈ N, 4 \lt n \lt 10\Big\rbrace$.

n can be 5, 6, 7, 8, 9.

Calculating x = $\dfrac{1}{n}$:

For n = 5, $x = \dfrac{1}{5}$

For n = 6, $x = \dfrac{1}{6}$

For n = 7, $x = \dfrac{1}{7}$

For n = 8, $x = \dfrac{1}{8}$

For n = 9, $x = \dfrac{1}{9}$

$J = \Big\lbrace\dfrac{1}{5}, \dfrac{1}{6}, \dfrac{1}{7}, \dfrac{1}{8}, \dfrac{1}{9}\Big\rbrace$, n(J) = 5

Write each of the following sets in set-builder form :

(i) A = {4, 6, 8, 9, 10, 12, 14, 15, 16, 18}.

(ii) B = {1, 2, 3, 5, 6, 10, 15, 30}.

(iii) C = {-9, -6, -3, 0, 3, 6, 9, 12, 15}.

(iv) D = $\Big\lbrace\dfrac{1}{2},\dfrac{2}{3},\dfrac{3}{4},...............,\dfrac{8}{9}\Big\rbrace$.

(v) E = $\Big\lbrace\dfrac{1}{3},\dfrac{1}{5},\dfrac{1}{7},\dfrac{1}{11},\dfrac{1}{13},\dfrac{1}{17},\dfrac{1}{19}\Big\rbrace$.

(vi) F = {April, June, September, November}.

(vii) G = {0}.

(viii) H = { }.

Answer

(i) A = {4, 6, 8, 9, 10, 12, 14, 15, 16, 18}.

These are composite numbers (numbers with more than two factors) less than 20.

∴ A = {x : x is a composite number, 1 < x < 20}

(ii) B = {1, 2, 3, 5, 6, 10, 15, 30}.

These are all the natural numbers that divide 30 without a remainder.

∴ B = {x : x is a factor of 30}

(iii) C = {-9, -6, -3, 0, 3, 6, 9, 12, 15}.

These are multiples of 3 ranging from -9 to 15.

∴ C = {x : x = 3n, n ∈ I, -3 ≤ n ≤ 5}

(iv) D = $\Big\lbrace\dfrac{1}{2},\dfrac{2}{3},\dfrac{3}{4},...............,\dfrac{8}{9}\Big\rbrace$.

Each element is a fraction where the denominator is 1 more than the numerator $\Big(\dfrac{n}{n+1}\Big)$, and the numerator n is a natural number from 1 to 8.

∴ D = {x : x = $\dfrac{n}{n+1}$, n ∈ N and 1 ≤ n ≤ 8}

(v) E = $\Big\lbrace\dfrac{1}{3},\dfrac{1}{5},\dfrac{1}{7},\dfrac{1}{11},\dfrac{1}{13},\dfrac{1}{17},\dfrac{1}{19}\Big\rbrace$.

The denominators are prime numbers between 2 and 20.

∴ E = {x : x = $\dfrac{1}{n}$, n is prime, 2 < n < 20}

(vi) F = {April, June, September, November}.

These are the specific months of the year that contain exactly 30 days.

∴ F = {x : x is a month of the year having 30 days}

(vii) G = {0}.

Zero is the only whole number that is not a natural number.

∴ G = {x : x + 1 = 1, x ∈ W}

(viii) H = { }.

An empty set contains no elements. We can describe it using a property that is impossible to satisfy.

∴ H = {x : x is a number, x ≠ x}

State whether the given set is finite or infinite :

(i) Set of all even natural numbers.

(ii) Set of all odd integers.

(iii) Set of all rivers in India.

(iv) Set of all points on a line segment 1 cm long.

(v) Set of all factors of 1200.

(vi) Set of all multiples of 6.

(vii) Set of all drops of water in a bucket.

Answer

(i) Set of all even natural numbers.

Natural numbers go on forever (2, 4, 6, 8, .....), so the counting process never ends.

∴ It is an infinite set.

(ii) Set of all odd integers.

Integers extend infinitely in both positive and negative directions (...., -3, -1, 1, 3, ....).

∴ It is an infinite set.

(iii) Set of all rivers in India.

Although there are many rivers, the total number is a specific, countable figure that can be listed.

∴ It is a finite set.

(iv) Set of all points on a line segment 1 cm long.

Mathematically, a line segment consists of an uncountable infinity of points, regardless of its length.

∴ It is an infinite set.

(v) Set of all factors of 1200.

Every number has a limited (finite) number of factors.

∴ It is a finite set.

(vi) Set of all multiples of 6.

Multiples are generated by multiplying 6 by natural numbers (6, 12, 18, ....), which are endless.

∴ It is an infinite set.

(vii) Set of all drops of water in a bucket.

The number is extremely large and difficult to count.

∴ It is an infinite set.

Identify the null sets among the following :

(i) A = {x : x is a whole number, x < 1}.

(ii) B = {x : x is a number, x > x}.

(iii) C = {x : x is an even prime number}.

(iv) D = {x : x ∈ I, x² = -4}.

(v) E = {x : x is a perfect square number, 40 < x < 50}.

(vi) F = {x : x ∈ N, 5 < x < 6}.

Answer

(i) A = {x : x is a whole number, x < 1}.

Whole numbers start from 0. The only whole number less than 1 is 0.

A = {0}, it contains one element.

∴ It is not a null set.

(ii) B = {x : x is a number, x > x}.

Mathematically, no number can be strictly greater than itself.

∴ It is a null set.

(iii) C = {x : x is an even prime number}.

The number 2 is the only even prime number.

C = {2}

∴ It is not a null set.

(iv) D = {x : x ∈ I, x² = -4}.

The square of any integer (I) is always non-negative (e.g., 2² = 4 and (-2)² = 4). There is no integer whose square is -4.

D = { }

∴ It is a null set.

(v) E = {x : x is a perfect square number, 40 < x < 50}.

Perfect squares near this range are 6² = 36 and 7² = 49. Since 49 falls between 40 and 50, the set contains an element.

E = {49}

∴ It is not a null set.

(vi) F = {x : x ∈ N, 5 < x < 6}.

Natural numbers (N) are counting numbers (1, 2, 3, .....). There is no natural number between 5 and 6.

F = { }

∴ It is a null set.

Identify whether the given pair consists of equal or equivalent but not equal sets or none :

(i) A = set of letters of the word 'FLOWER'.
    B = set of letters of the word 'FOLLOWER'.

(ii) C = {x : x ∈ N, x + 5 = 6} and D = {x : x ∈ W, x < 1}.

(iii) E = set of first five whole numbers.
      F = set of first five natural numbers.

(iv) G = {a, b, c} and H = {x, y, z}.

(v) J = {x : x ∈ N, x ≠ x} and K = {x : x ∈ N, 6 < x < 7}.

Answer

(i)

A = Set of letters of the word 'FLOWER'.
B = Set of letters of the word 'FOLLOWER'.

A = {F, L, O, W, E, R} and B = {F, O, L, W, E, R}

In roster form, repeated letters in 'FOLLOWER' are listed only once.

Both sets contain the same elements {F, L, O, W, E, R}. These are equal sets.

∴ A = B

(ii) C = {x : x ∈ N, x + 5 = 6} and D = {x : x ∈ W, x < 1}.

For Set C:

x + 5 = 6

⇒ x = 6 - 5

⇒ x = 1

Since 1 is a natural number, x = {1}.

C = {1}, n(C) = 1

For Set D:

x < 1

The only whole number less than 1 is 0.

D = {0}, n(D) = 1

n(C) = 1 and n(D) = 1. They have the same number of elements but different elements.

They are equivalent but not equal sets

∴ C ↔ D

(iii)

E = set of first five whole numbers.
F = set of first five natural numbers.

E = {0, 1, 2, 3, 4} and F = {1, 2, 3, 4, 5}

E contains 0 to 4 (first five whole numbers). F contains 1 to 5 (first five natural numbers).

n(E) = 5 and n(F) = 5. The cardinal numbers are the same, but the elements are different.

They are equivalent but not equal sets

∴ E ↔ F

(iv) G = {a, b, c} and H = {x, y, z}.

Both sets have three distinct elements.

n(G) = 3 and n(H) = 3. The elements are entirely different.

They are equivalent but not equal sets

∴ G ↔ H

(v) J = {x : x ∈ N, x ≠ x} and K = {x : x ∈ N, 6 < x < 7}.

No natural number is unequal to itself. J = { } (Null set).

There is no natural number between 6 and 7. K = { } (Null set).

Both are empty sets and therefore contain exactly the same (zero) elements. These are equal sets.

∴ J = K

For each of the following pairs of sets, identify the disjoint and overlapping sets :

(i)

A = {x : x is a prime number, x < 8}.
B = {x : x is an even natural number, x < 8}.

(ii) C = {x : x ∈ N, x < 10} and D = {x : x ∈ N, x is a multiple of 5}.

(iii) E = {x : x = 4n, n ∈ N} and F = {x : x = 9n, n ∈ N}.

(iv) G = {x : x = 8n, n ∈ N and n < 7} and H = {x : x = 9n, n ∈ N and n < 7}.

Answer

(i)

A = {x : x is a prime number, x < 8}.
B = {x : x is an even natural number, x < 8}.

Roster Form of A = {2, 3, 5, 7}

Roster Form of B = {2, 4, 6}

Both sets share the element {2}.

∴ These are overlapping sets

(ii) C = {x : x ∈ N, x < 10} and D = {x : x ∈ N, x is a multiple of 5}.

Roster Form of C = {1, 2, 3, 4, 5, 6, 7, 8, 9}

Roster Form of D = {5, 10, 15, 20, $\dots$}

The number 5 is present in both sets.

∴ These are overlapping sets

(iii) E = {x : x = 4n, n ∈ N} and F = {x : x = 9n, n ∈ N}.

Roster Form of E = {4, 8, 12, 16, 20, 24, 28, 32, 36, ....}

Roster Form of F = {9, 18, 27, 36, 45, ....}

The first common multiple is 36. Since these are infinite sets of multiples, they will continue to have common elements (like 72, 108, etc.).

∴ These are overlapping sets

(iv) G = {x : x = 8n, n ∈ N and n < 7} and H = {x : x = 9n, n ∈ N and n < 7}.

Roster Form of G = {8, 16, 24, 32, 40, 48} (Multiples of 8 up to n=6)

Roster Form of H = {9, 18, 27, 36, 45, 54} (Multiples of 9 up to n=6)

There are no common elements in these finite lists. The first shared multiple of 8 and 9 is 72, which is outside the range of both sets.

∴ These are disjoint sets

State in each case, whether the given statement is true or false :

(i) If A is the set of all non-negative integers, then 0 ∈ A.

(ii) If B is the set of all consonants, then c ∈ B.

(iii) If C is the set of all prime numbers less than 80, then 57 ∈ C.

(iv) {x : x ∈ W, x + 5 = 5} is a singleton set.

(v) If D = {x : x ∈ W, x < 4}, then n(D) = 4.

(vi) {a, b, c, 1, 2, 3} is not a set.

(vii) {1, 2, 3, 1, 2, 3, 1, 2, 3,...............} is an infinite set.

(viii) 0 ∈ Φ.

(ix) {3, 5} ∈ (1, 3, 5, 7, 9).

Answer

(i) True
Reason — Non-negative integers include all natural numbers and zero (0, 1, 2, 3, ....). Therefore, 0 is an element of set A.

(ii) True
Reason — Consonants are all letters of the alphabet except for vowels (a, e, i, o, u). Since 'c' is not a vowel, it belongs to the set of consonants B.

(iii) False
Reason — A prime number has only two factors: 1 and itself. Since 57 is divisible by 3 (3 x 19 = 57), it is a composite number, not a prime number.

(iv) True
Reason — Solving x + 5 = 5 gives x = 0. Since 0 is a whole number (W), the set contains exactly one element: {0}. A set with one element is called a singleton set.

(v) True
Reason — The whole numbers (W) less than 4 are {0, 1, 2, 3}. Counting these elements gives a cardinal number n(D) = 4.

(vi) False
Reason — A set can contain any well-defined collection of distinct objects, including a mix of letters and numbers. Therefore, {a, b, c, 1, 2, 3} is a valid set.

(vii) False
Reason — In set theory, repeated elements are counted only once. The set {1, 2, 3, 1, 2, 3, ....} contains only the distinct elements {1, 2, 3}. Since it has a limited number of distinct elements, it is a finite set, not an infinite one.

(viii) False
Reason — The symbol Φ represents a null set, which by definition contains no elements at all. Therefore, 0 cannot be an element of Φ.

(ix) False
Reason — The notation {3, 5} represents a subset, not an element. The correct notation would be {3, 5} ⊂ {1, 3, 5, 7, 9} or 3, 5 ∈ {1, 3, 5, 7, 9}.