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Mathematics · Class 6 · STBB

Unit 1 — Sets

Sindh Textbook Board — Solved notes with verified answers

Key Concepts ✓  Exercise 1.1 ✓  Exercise 1.2 ✓  Review Exercise 1 ✓

1. Key Concepts — What is a Set?

• Georg Cantor, a German mathematician, was the first person who gave the concept of a set in the 19th century. He is called the father of set theory.
• Set: A collection of “well-defined” and “distinct” (different) objects.
• Each object of a set is called an element or member of the set.
• Well-defined means we can clearly say whether an object belongs to the set or not.
• Distinct means no element is repeated in a set.

Points to Remember

• The name of a set is always denoted by capital letters of the English alphabet, e.g. A, B, C, P, Q.
• Every element of a set is separated from the next by a comma (,).
• All the members of a set are written inside curly brackets { }. This form is called the tabular form of a set.
• The order of writing elements does not matter in a set. E.g. {1, 2, 3} and {3, 2, 1} are the same set.
• The elements of a set are not repeated.
• The symbol ∈ (∈) means “belongs to” or “is an element of” a set.
• The symbol ∉ (∉) means “does not belong to” a set.

Example: If A = {2, 4, 6}  then  2 ∈ A,  4 ∈ A,  but  3 ∉ A.

2. Exercise 1.1 — Solved

Q1. Tick (✓) which of the following are sets:

(i) Collection of the names of Presidents of Pakistan  — Set ✓
(ii) Collection of names of captains of hockey teams of Pakistan  — Set ✓
(iii) Collection of delicious dishes  — Not a set ✗ (not well-defined)
(iv) Collection of intelligent students in your class  — Not a set ✗ (intelligent is not well-defined)
(v) Collection of greater numbers  — Not a set ✗ (not well-defined)
(vi) Collection of English teachers in your school  — Set ✓

Reason: A good collection is a set only when it is well-defined and we can clearly decide which objects belong to it.

Q2. If A = {5, 6, 7, 8} and B = {a, b, c, d}, fill in the blanks with the symbols ∈ or ∉:

(i) 5 ∈ A
(ii) 9 ∉ A
(iii) m ∉ A
(iv) c ∈ B
(v) b ∈ B
(vi) 7 ∉ B

Q3. Write the following sets in tabular form:

(i) Set of names of capitals of the provinces of Pakistan.

Ans: {Karachi, Lahore, Peshawar, Quetta, Gilgit}

(ii) Set of natural numbers from 50 to 70.

Ans: {50, 51, 52, 53, …, 69, 70}

(iii) Set of first ten natural numbers divisible by 2.

Ans: {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}

(iv) Set of letters in the word “PAKISTAN”.

Ans: {P, A, K, I, S, T, N}  (repeated letters are written only once)

(v) Set of mathematics teachers in your school.

Ans: Example: {Miss Iqra, Sir Saqib}  (answer may vary from school to school)

(vi) Set of colors of the national flag.

Ans: {White, Green}

Q4. If P = {a, e, i, o, u} and Q = {1, 2, 3, …, 10}, then which of the following statements are true or false?

(i) a ∈ P  — True
(ii) 1 ∈ Q  — True
(iii) i ∈ Q  — False
(iv) v ∈ P  — False
(v) y ∈ Q  — False
(vi) e ∈ P  — True
(vii) 3 ∈ P  — False
(viii) 7 ∈ Q  — True

3. Exercise 1.2 — Solved

Key Concepts of Exercise 1.2

• Finite set: A set which has a limited (countable) number of elements. E.g. {2, 4, 6, 8}
• Infinite set: A set which has an unlimited (uncountable) number of elements. E.g. {1, 2, 3, 4, … }
• Empty set: A set which has no element. It is also called null or void set. It is denoted by Ø or { }.
• Singleton set: A set having only one element. E.g. {5}
• Equal sets: Two sets are equal if all the elements of one set are the same as the elements of the other. E.g. {1, 2, 3} = {2, 3, 1}
• Equivalent sets: Two sets having the same number of elements. E.g. A = {1, 2, 3} and B = {a, b, c} (both have 3 elements).
• Subset: If each element of set A is also an element of set B, then A is called a subset of B. Symbolically we write A ⊆ B.
• Every set is a subset of itself.
• Superset: If A is a subset of B, then B is called a superset of A. Symbolically we write B ⊇ A.
• Proper subset: If A is a subset of B but A is not equal to B, then A is called a proper subset of B. Symbolically we write A ⊂ B.
• Improper subset: If A is a subset of B and A is also equal to B, then A is called an improper subset of B.

Q1. Give reasons why the following collections are not sets:

(i)  {d, o, o, r}

Ans: “o” is repeating, so it is not a set.

(ii)  {«, O, «, D}

Ans: The element “«” is repeating, so it is not a set.

(iii)  {a, f, d, a}

Ans: “a” is repeating, so it is not a set.

(iv)  {2, 2, 3, 3, 4, 4}

Ans: “2”, “3” and “4” are repeating, so it is not a set.

(v)  Set of beautiful birds

Ans: Not a set — the word “beautiful” is not well-defined.

(vi)  Set of good players

Ans: Not a set — the word “good” is not well-defined.

Q2. Which of the following are finite sets or infinite sets?

(i) A = {0, 1, 2, …, 50}  — Finite set
(ii) B = {100, 200, 300, 400, …}  — Infinite set
(iii) Set of hair on the body of a goat  — Infinite set
(iv) Set of legs of a cat  — Finite set
(v) Set of stars in the sky  — Finite set
(vi) Set of lines passing through a point  — Infinite set
(vii) Set of natural numbers greater than 20  — Infinite set
(viii) Set of all the cities of Pakistan  — Finite set
(ix) Set of all the schools in Sindh  — Finite set
(x) Set of all even numbers  — Infinite set

Q3. Which of the following are empty sets?

(i) Set of students of your class over 20 years of age.

Ans: Empty set (no student in a class is over 20 years old)

(ii) Set of letters after “Z” in the English alphabet.

Ans: Empty set (there is no letter after Z)

(iii) Set of children whose names start with letter “K” in your locality.

Ans: Not necessarily an empty set (depends on your locality)

(iv) Set of names of the solar calendar months starting with the letter “Z”.

Ans: Empty set (no month starts with Z)

(v) Set of fishes which live in sand.

Ans: Empty set

(vi) Set of even numbers between 4 and 10.

Ans: {6, 8} — not an empty set

(vii) {0}

Ans: Not an empty set (it contains the element 0 — it is a singleton set)

(viii) {Ø}

Ans: Not an empty set (it contains one element — the symbol Ø itself)

Q4. Which pairs of the following are equal sets?

(i) {1, 2, 3} and {2, 3, 1}  — Equal sets ✓
(ii) {p, a, t} and {t, a, p}  — Equal sets ✓
(iii) {k, i, t, e} and {b, i, t, e}  — Not equal (k ≠ b)
(iv) {x, y, z} and the set of first three letters of the English alphabet  — Not equal ({x, y, z} ≠ {a, b, c})
(v) Set of odd numbers less than 2 and { }  — Not equal ({1} ≠ { })

Answer key: (i) and (ii) are equal sets.

Q5. Which pairs of the following are equivalent sets?

(i) {m, i, l, e} and {n, i, l, e}  — Equivalent sets (both have 4 elements)
(ii) {1, 2, 3} and {3, 2, 1}  — Equivalent sets (both have 3 elements)
(iii) {a, b, c, d, e} and {1, 2, 3, 4, 5}  — Equivalent sets (both have 5 elements)
(iv) {6, 66, 666, 6666} and {666, 7777, 77}  — Not equivalent (4 ≠ 3 elements)

Answer key: (i), (ii) and (iii) are equivalent sets. Remember: Equal sets are always equivalent, but equivalent sets may not be equal.

Q6. Which of the following statements are true or false?

(i) B = {m, o, n}, if B is the set of letters of the word “MOON”  — True
(ii) If A = {1, 2} and B = {2, 3, 4, 5}, then A ⊂ B  — False (1 is not an element of B)
(iii) If A = {4, 5, 10} and B = {5, 10, 20}, then A ∼ B  — True (both are equivalent, having 3 elements)
(iv) If A = {0, 1, 2, 3, 5} and B is the set of natural numbers less than 6, then A = B  — False
(v) If X is equal to Y, then X is equivalent to Y  — True

Q7. Choose the appropriate symbol ( = , ⊆ , ¬⊂ , ⊇ , ∼ , ⊂ ) to indicate the relation between the following sets:

(i) {10, 20, 30} ⊆ {5, 10, 15, 20, 25, 30}
(ii) {7, 14, 21, 28, 35, 42, 49} = set of multiples of 7 less than 50
(iii) {5, 6, 7, 8, …} ⊇ {7, 5, 8}
(iv) {0} ⊇ { }
(v) {t, e, a} ∼ {a, e, t}  (they are also equal sets)
(vi) {Hyderabad, Karachi, Sukkur} ⊆ set of all the cities of Sindh
(vii) {a, e, i, o, u} ⊂ set of letters of the English alphabet
(viii) {11, 22, 33} ⊆ {1, 2, 3, …}

Answer key: (i) ⊆, (ii) =, (iii) ⊇, (iv) ⊇, (v) ∼, (vi) ⊆, (vii) ⊂, (viii) ⊆

4. Review Exercise 1 — Solved

Q1. Fill in the blanks:

(i) If A = {a, b, c}, then a, b, c are elements (members) of set A.
(ii) A set which has no element is called an empty (null/void) set.
(iii) A set which has a limited number of elements is called a finite set.
(iv) If two sets A and B have all the same elements, then they are said to be equal sets.

Q2. Choose the correct answer:

(i) If B is the set of vowels of the English alphabet: (a) p ∈ B    (b) e ∉ B    (c) b ∈ B    (d) a ∈ B
   Answer: (d) a ∈ B

(ii) { } is called: (a) Null set    (b) Infinite set    (c) Subset    (d) Singleton set
   Answer: (a) Null (empty) set  Note: the answer key in the book prints “(d)”, but that is a printing error — a set with no element is the null/empty set, not a singleton set.

(iii) If sets A and B are equal, we use the symbol: (a) ∈    (b) ⊂    (c) ∼    (d) =
   Answer: (d) = (A = B)

(iv) If A = {a, b, c} and B = {a, b, c, d, e}, then: (a) A = B    (b) A ∼ B    (c) A ⊂ B    (d) B ⊂ A
   Answer: (c) A ⊂ B (A is a proper subset of B)

Q3. Write two examples of a set:

Ans: (1) Set of even numbers between 1 and 10 = {2, 4, 6, 8}
(2) Set of vowels in the English alphabet = {a, e, i, o, u}

Q4. Write the following sets in tabular form:

(i) A is the set of all even numbers between 2 and 10.

Ans: A = {4, 6, 8}

(ii) B is the set of all odd numbers from 1 to 17.

Ans: B = {1, 3, 5, 7, 9, 11, 13, 15, 17}

(iii) C is the set of natural numbers divisible by 2 and less than 30.

Ans: C = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28}

(iv) D is the set of months of a solar calendar beginning with the letter “J”.

Ans: D = {January, June, July}

Q5. Which of these sets are finite and which are infinite?

(i) A is the set of members of your family  — Finite set
(ii) B is the set of all even numbers  — Infinite set
(iii) C is the set of prime factors of 60  — Finite set (C = {2, 3, 5})
(iv) D is the set of all multiples of 7  — Infinite set
(v) E is the set of one-digit natural numbers  — Finite set (E = {1, 2, 3, …, 9})

Q6. Which of the following are empty sets?

(i) L is the set of names of the days of the week which start with the letter “S”.

Ans: L = {Saturday, Sunday} — not an empty set

(ii) F is the name of a month of a year having 32 days.

Ans: Empty set

(iii) M is the set of odd numbers exactly divisible by 2.

Ans: Empty set (odd numbers are never divisible by 2)

(iv) N is the set of prime numbers exactly divisible by 2.

Ans: N = {2} — not an empty set

Q7. If A = {d}, B = {c, d}, C = {a, b, c}, D = {a, b}, then which of the following statements are true or false?

(i) D ⊆ C  — True
(ii) A ¬⊂ C  — True (d is not an element of C)
(iii) B ⊆ D  — False (c and d are not elements of D)
(iv) C ¬⊂ D  — True (c is not an element of D)

Q8. Indicate which of the following statements are true or false:

(i) The members of any set are always of the same kind  — True
(ii) The order of the elements of a set does not matter  — True
(iii) An object can be included in a set repeatedly  — False
(iv) Any collection of objects is called a set  — False
(v) The notation “∈” is a symbol to indicate membership of a set  — True
(vi) A well-defined collection of distinct objects is called a set  — True
(vii) {a, b, c, d} is not a set  — False
(viii) The notation “∉” is a symbol to indicate membership of a set  — False (∉ means “does not belong to”; ∈ is the symbol of membership)
(ix) 1, 2, 3, 4 is a set  — False (it should be written in curly brackets {1, 2, 3, 4})

5. Quick Formula / Symbol Card

• ∈  belongs to     ∉  does not belong to
• ⊂  proper subset     ⊆  subset / improper subset
• ⊃  proper superset     ⊇  superset
• Ø or { }  empty (null) set
• =  equal sets (same elements)     ∼  equivalent sets (same number of elements)
• ≠  not equal     n(A)  number of elements (cardinality) of set A

Notes based on Sindh Textbook Board (STBB) – Mathematics for Class 6, Unit 1: Sets. Solvable answers that depend on real life (e.g. teachers in your school / children in your locality) are given as examples.
Prepared for study & revision purpose only.