# Sets

Chapters

- ap-scert-10th-class-maths-lesson-1-Real-Numbers
- ap-scert-10th-class-Maths-Lesson-2-Sets
- ap-scert-10th-class-Maths-Lesson-3-Polynomials
- ap-scert-10th-class-Maths-Lesson-4-Pair-of-Linear-Equations-in-Two-Variables
- ap-scert-10th-class-Maths-Lesson-5-Quadratic-Equations
- ap-scert-10th-class-Maths-Lesson-6-Progressions
- ap-scert-10th-class-Maths-Lesson-7-Coordinate-Geometry
- ap-scert-10th-class-Maths-Lesson-8-Similar-Triangles
- ap-scert-10th-class-Maths-Lesson-9-Tangents-and-Secants-to-a-Circle
- ap-scert-10th-class-Maths-Lesson-10-Mensuration
- ap-scert-10th-class-Maths-Lesson-11-Trigonometry
- ap-scert-10th-class-Maths-Lesson-12-Applications-of-Trigonometry
- ap-scert-10th-class-Maths-Lesson-13-Probability
- ap-scert-10th-class-Maths-Lesson-14-Statistics

- ap-scert-10th-class-maths-lesson-1-Real-Numbers
- ap-scert-10th-class-Maths-Lesson-2-Sets
- ap-scert-10th-class-Maths-Lesson-3-Polynomials
- ap-scert-10th-class-Maths-Lesson-4-Pair-of-Linear-Equations-in-Two-Variables
- ap-scert-10th-class-Maths-Lesson-5-Quadratic-Equations
- ap-scert-10th-class-Maths-Lesson-6-Progressions
- ap-scert-10th-class-Maths-Lesson-7-Coordinate-Geometry
- ap-scert-10th-class-Maths-Lesson-8-Similar-Triangles
- ap-scert-10th-class-Maths-Lesson-9-Tangents-and-Secants-to-a-Circle
- ap-scert-10th-class-Maths-Lesson-10-Mensuration
- ap-scert-10th-class-Maths-Lesson-11-Trigonometry
- ap-scert-10th-class-Maths-Lesson-12-Applications-of-Trigonometry
- ap-scert-10th-class-Maths-Lesson-13-Probability
- ap-scert-10th-class-Maths-Lesson-14-Statistics

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Solutions

Which of the following are sets? Justify your answer.

- i) The collection of all the months of a year beginning with die letter “J”.
- ii) The collection of ten most talented writers of India.
- iii) A team of eleven best cricket batsmen of the world.
- iv) The collection of all boys in your class.
- v) The collection of all even integers.

Answer:

- i) There are 3 months as January, June and July beginning with letter ‘J’. Therefore, it is a well defined collection of months and hence it is a set.
- ii) The concept of talented writers of India is vague, since there is no rule given for deciding whether a particular writer is talented or not.
- Hence, the given collection is not a set.
- iii) A team of eleven best cricket batsmen of the world is vague, since there is no rule given for deciding whether a particular batsman is the best.
- Hence, the given collection is not a set.
- iv) The collection of all boys in my class is well defined. Hence, the given collection is a set.
- v) The collection of all even integers i.e., (2, 4, 6, 8, ……) is well defined.
- Hence, the given collection is a set.

If A = {0, 2, 4, 6}, B = {3, 5, 7} and C = {p, q, r} then fill the appropriate symbol, ∈ or ∉ in the blanks.

- i) 0 ….. A
- ii) 3 ….. C
- iii) 4 ….. B
- iv) 8 ….. A
- v) p ….. C
- vi) 7 ….. B

Answer:

- i) ∈
- ii) ∉
- iii) ∉
- iv) ∉
- v) ∈
- vi) ∈

Express the following statements using symbols.

- i) The elements ‘x’ does not belong to ‘A’.
- ii) ‘d’ is an element of the set ‘B’.
- iii) ‘1’ belongs to the set of Natural numbers N.
- iv) ‘8′ does not belong to the set of prime numbers P.

Answer:

- i) x ∉ A
- ii) d ∈ B
- iii) 1 ∈ N
- iv) 8 ∉ P

State whether the following statements are true or false. Justify your answer.

- i) 5 ∉ set of prime numbers
- ii) S = {5, 6, 7} implies 8 ∈ S.
- iii) -5 ∉ W where ‘W’ is the set of whole numbers.
- iv)8/11 ∈ Z

where ‘Z’ is the set of integers.

Answer:

- i) False
- ii) False
- iii) True
- iv) False

Write the following sets in roster form.

- i) B = {x : x is a natural number smaller than 6}.
- ii) C = {x : x is a two-digit natural number such that the sum of its digits is 8}.
- iii) D = {x : x is a prime number which is a divisor of 60}.
- iv) E = {x : x is an alphabet in BETTER}.

Answer:

- i) B = {1, 2, 3, 4, 5}
- ii) C = {17, 26, 35, 44, 53, 62, 71}
- iii) D = {5, 3}
- iv) E = {B, E, T, R}

Write the following sets in the set – builder form.

- i) {3, 6, 9, 12}
- ii) {2, 4, 8, 16, 32}
- iii) {5, 25, 125, 625}
- iv) {1, 4, 9, 16, 25, …, 100}

Answer:

- i) A = {x : x is multiple of 3 and less than 13}
- ii) B = {x : x = 2
^{P}, 0 < P < 6, P ∈ N} - iii) C = {x : x = 5
^{P}, 0 < P < 5, P ∈ N} - iv) D = {x : x = P
^{2}, 0 < P < 11, P ∈ W}

Write the following sets in roster form.

- i) A = {x: x is a natural number greater than 50 but smaller than 100}
- ii) B = {x : x is an integer, x
^{2}= 4} - iii) D = {x : x is a letter in the word “LOYAL”}

Answer:

- i) A = {51, 52, 53, ……. , 98, 99}
- ii) B = {+2, -2}
- iii) D = {L, O, Y, A}

Match the roster form with set builder form.

i) {1, 2, 3, 6} ( ) a) {x : x is a prime number and a divisor of 6}

ii) {2, 3} ( ) b) {x : x is an odd natural number smaller than 10}

iii) {M, A, T, H, E, I, C, S} ( ) c) {x : x is a natural number and divisor of 6}

iv) {1, 3, 5, 7, 9} ( ) d) {x : x is a letter of the word MATHEMATICS}

Answer:

- i) c
- ii) a
- iii) d
- iv) b

If A = {1, 2, 3, 4}; B = {1, 2, 3, 5, 6} then find A ∩ B and B ∩ A. Are they equal ?

Answer:

Given sets are A = {1, 2, 3, 4} and B = {1,2,3, 5,6}

A ∩ B = {1,2, 3,4} ∩ {1,2, 3, 5, 6}

= {1,2,3} …… (1)

B ∩ A = {1, 2, 3, 5, 6} ∩ {1, 2, 3, 4}

= {1,2,3} …….(2)

From (1) and (2)

A ∩ B and B ∩ A are same.

A = {0, 2, 4}, find A ∩ ϕ and A ∩ A. Comment.

Answer:

Given set A = {0, 2, 4} and ϕ is a null set.

A ∩ ϕ = {0, 2, 4} ∩ { }

= { } ……. (1)

A ∩ A = {0, 2, 4} ∩ {0, 2, 4}

= {0, 2,4} …….. (2)

From (1) and (2),

We conclude that A ∩ ϕ = ϕ and A ∩ A = A

If A = {2, 4, 6, 8, 10} and B = {3, 6, 9, 12, 15}, find A - B and B - A.

Answer:

Given sets are

A {2, 4, 6, 8, 10} and B = {3, 6, 9, 12, 15}

A - B = {2, 4, 6, 8, 10} - {3, 6, 9, 12, 15}

= {2, 4, 8, 10} …… (1)

B – A = {3, 6, 9, 12, 15} - {2, 4, 6, 8, 10}

= {3, 9, 12, 15} …… (2)

From (1) and (2), A - B ≠ B - A

If A and B are two sets such that A ⊂ B then, what is A ∪ B?

Answer:

Let us consider A ⊂ B

Set A = {1, 2, 3} and

Set B = {1, 2, 3, 4, 5}

Now A ∪ B = {1, 2, 3} ∪ {1, 2, 3, 4, 5}

= {1, 2, 3, 4, 5} = B

∴ A ∪ B = B

If A = {x : x is a natural number},

B = {x : x is an even natural number},

C = {x : x is an odd natural number} and

D = {x : x is a prime number}

Find A ∩ B, A ∩ C, A ∩ D, B ∩ C, B ∩ D, C ∩ D.

Answer:

Given sets are

A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ……}

B = {2, 4, 6, 8, 10, …….}

C = {1, 3, 5, 7, 9, …….}

D = {2, 3, 5, 7, 11, …….}

A ∩ B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …….} ∩ {2, 4, 6, 8, 10, ……}

= {2, 4, 6, 8, 10, ……}

A ∩ C = {1, 2, 3,4, 5, 6, 7, 8, 9, 10, …} ∩ {1, 3, 5, 7, 9 }

= {1, 3, 5, 7, 9, ……}

A ∩ D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} ∩ {2, 3, -5, 7, 11,….}

= {2, 3, 5, 7, 11, ……}

B ∩ C = {2, 4, 6, 8, 10, ……} ∩ {1, 3, 5, 7, 9, …….}

= { } = ϕ

B ∩ D = {2, 4, 6, 8, 10, ……} ∩ {2, 3, 5, 7, 11, ……}

= {2}

C ∩ D = {1, 3, 5, 7, 9, ……} ∩ {2, 3, 5, 7, 11, 13, ……}

= {3, 5, 7, …..}

If A = {3, 6, 9, 12, 15, 18, 21}; B = {4, 8, 12, 16, 20}; C = {2, 4, 6, 8, 10, 12, 14, 16}; D = {5, 10, 15, 20} find

- (i) A - B
- (ii) A - C
- (iii) A - D
- (iv) B - A
- (v) C - A
- (vi) D - A
- (vii) B - C
- (viii) B - D
- (ix) C - B
- (x) D - B

Answer:

Given sets are A = {3, 6, 9, 12, 15, 18, 21}

B = {4, 8, 12, 16, 20}

C = {2, 4, 6, 8, 10, 12, 14, 16} and

D = {5, 10, 15, 20}

- i) A - B = {3, 6, 9, 12, 15, 18, 21} - {4, 8, 12, 16, 20} = {3, 6, 9, 15, 18, 21}
- ii) A - C = {3, 6, 9, 12, 15, 18, 21} - {2, 4, 6, 8, 10, 12, 14, 16} = {3,9,15,18,21}
- iii) A - D = {3, 6, 9, 12, 15, 18, 21} - {5, 10, 15, 20} = {3,6,9,12,18,21}
- iv) B - A = {4, 8, 12, 16, 20} - {3, 6, 9, 12, 15, 18, 21} = {4, 8, 16, 20}
- v) C - A = {2, 4, 6, 8, 10, 12, 14, 16} - {3, 6, 9, 12, 15, 18, 21} = {2, 4, 8, 10, 14, 16}
- vi) D - A = {5, 10, 15, 20} - {3, 6, 9, 12, 15, 18, 21} = {5, 10, 20}
- vii) B - C = {4, 8, 12, 16, 20} - {2, 4, 6, 8, 10, 12, 14, 16} = {20}
- viii) B - D = {4, 8, 12, 16,20} - {5, 10, 15, 20} = {4, 8, 12, 16}
- ix) C - B = {2, 4, 6, 8, 10, 12, 14, 16} - {4, 8, 12, 16, 20} = {2, 6, 10, 14}
- x) D - B = {5, 10, 15, 20} - {4, 8, 12, 16, 20} = {5, 10, 15}

State whether each of the following statement is true or false. Justify your answers.

- i) {2,3,4,5} and {3,6} are disjoint sets.
- ii) {a, e, i, o, u} and {a, b, c, d} are disjoint sets.
- iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.
- iv) {2, 6, 10} and {3, 7, 11} are disjoint sets.

Answer:

i) Rule: If two sets are disjoint their intersection is null set.

= {2, 3, 4, 5} n {3, 6} = { 3 } ≠ ϕ

∴ Given statement is False.

ii) Given sets are

{a, e, i, o, u} and {a, b, c, d}

= {a, e, i, o, u} ∩ {a, b, c, d}

= { a } ≠ ϕ

∴ Given statement is False.

iii) Given sets are

{2, 6, 10, 14} and {3, 7, 11, 15}

= {2, 6, 10, 14} ∩ {3, 7, 11, 15}

= { }

∴ Given statement is True.

iv) Given sets are

{2, 6, 10} and {3, 7, 11}

= {2, 6, 10} ∩ {3, 7, 11} = { }

∴ Given statement is True.

Which of the following sets are equal?

- i)A = {x : x is a letter in the word FOLLOW}
- ii) B = {x : x is a letter in the word FLOW}
- iii) C = {x : x is a letter in the word WOLF}

Answer:

- i) Elements in set A are {F, L, O, W}
- ii) Elements in set B are {F, L, O, W}
- iii) Elements in set C are {F, L, O, W} Sets A, B and C have same elements, Hence, they are equal sets.

Consider the following sets and fill up the blank in the statement given below with = or ≠ so as to make the statement true.

A = {1, 2, 3};

B = {The first three natural numbers};

C = {a, b, c, d};

D = {d, c, a, b};

E = {a, e, i, o, u};

F = {Set of vowels in English Alphabet}

- i) A …. B
- ii) A …. E
- iii) C …. D
- iv) D …. F
- v) F …. A
- vi) D …. E
- vii) F …. B

Answer:

- i) A = B
- ii) A ≠ E
- iii) C = D
- iv) D ≠ F
- v) F ≠ A
- vi) D ≠ E
- vii) F ≠ B

In each of the following, state whether A = B or not.

- i) A = {a, b, c, d} ; B = {d, c, a, b}
- ii) A = {4, 8, 12, 16} ; B = {8, 4, 16, 18}
- iii) A = {2, 4, 6, 8, 10}; B = {x : x is a positive even integer and x ≤ 10}
- iv) A = {x : x is a multiple of 10}; B = {10, 15, 20, 25, 30, …}

Answer:

- i) A = B
- ii) A ≠ B
- iii) A ≠ B
- iv) A ≠ B

State the reasons for the following :

i) {1, 2, 3, …., 10} ≠ {x : x ∈ N and 1 < x < 10}

ii) {2, 4, 6, 8, 10} ≠ {x : x = 2n+1 and x ∈ N}

iii) {5, 15, 30, 45} ≠ {x : x is a multiple of 15}

iv) {2, 3, 5, 7, 9} ≠ {x : x is a prime number}

Answer:

i) In R.H.S ‘x’ is greater than 1 and less than 10 but L.H.S is having both 1 and 10.

ii) L.H.S ≠ R.H.S

R.H.S: x = 2n + 1 is definition of odd numbers.

L.H.S: Given set is even numbers set.

iii) x is a multiple of 15.

So 5 does not exist.

iv) x is a prime number but 9 is not a prime number.

List all the subsets of the following sets.

- i) B = {p, q}
- ii) C = {x, y, z}
- iii) D = {a, b, c, d}
- iv) E = {1, 4, 9, 16}
- v) F = {10, 100, 1000}

Answer:

- i) Subsets of ‘B’ are {p}, {q}, {p, q}, ϕ
- ii) Subsets of ‘C’ are {x}, {y} {z}, {x, y}, {y, z}, {z, x}, {x, y, z} and ϕ (2
^{3}= 8) - iii) Subsets of ‘D’ are {a}, {b}, {c}, {d}, {a,b}, {b,c}, {c, d}, {a, c}, {a, d}, {b, d}, {a, b, c}, {b, c, d}, {a, b, d}, {a, c, d}, {a, b, c, d} and φ
- iv) Subsets of ‘E’ are
- ϕ, {1}, {4}, {9}, {16}, {1,4}, {1,9}, {1, 16}, {4, 9}, {4, 16}, (9, 16}, {1, 4, 9}, {1, 9, 16}, {4, 9, 16}, {1, 4, 16}, {1, 4, 9, 16}
- v) Subsets of ‘F’ are
- ϕ, {10}, {100}, {1000}, {10, 100}, {100, 1000}, {10, 1000}, {10, 100, 1000}.

State which of the following sets are empty and which are not?

- i) The set of lines passing through a given point.
- ii) Set of odd natural numbers divisible by 2.
- iii) {x : x is a natural number, x < 5 and x > 7}
- iv) {x: x is a common point to any two parallel lines}
- v) Set of even prime numbers.

Answer:

- i) Not empty
- ii) Empty
- iii) Empty
- iv) Empty
- v) Not empty

Which of the following sets are finite or infinite?

- i) The set of months in a year.
- ii) {1, 2, 3, …, 99, 100}
- iii) The set of prime numbers smaller than 99.

Answer:

- i) Finite
- ii) Finite
- iii) Finite

State whether each of the following sets is finite or infinite.

- i) The set of letters in the English alphabet.
- ii) The set of lines which are parallel to the X-axis.
- iii) The set of numbers which are multiples of 5.
- iv) The set of circles passing through the origin (0, 0).

Answer:

- i) Finite
- ii) Infinite
- iii) Infinite
- iv) Infinite

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