NCERT solutions for Mathematics [English] Class 11 chapter 1 - Sets [Latest edition]

Identify whether the following is set or not? Justify your answer.

The collection of all months of a year beginning with the letter J.

Identify whether the following is set or not? Justify your answer.

The collection of ten most talented writers of India.

Identify whether the following is set or not? Justify your answer.

A team of eleven best-cricket batsmen of the world.

Identify whether the following is set or not? Justify your answer.

The collection of all boys in your class.

Identify whether the following is set or not? Justify your answer.

The collection of all natural numbers less than 100.

Identify whether the following is set or not? Justify your answer.

A collection of novels written by the writer Munshi Prem Chand.

Identify whether the following is set or not? Justify your answer.

The collection of all even integers.

Identify whether the following is set or not? Justify your answer.

The collection of questions in this Chapter.

Identify whether the following is set or not? Justify your answer.

A collection of most dangerous animals of the world.

Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank space:

5 _____ A

Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank space:

8 ____ A

Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank space:

0 ____ A

Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank space:

4 _____ A

Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank space:

2 _____ A

Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank space:

10 _____ A

Write the following set in roster form:

A = {x : x is an integer and –3 ≤ x < 7}

Write the following set in roster form:

B = {x : x is a natural number less than 6}

Write the following set in roster form:

C = {x : x is a two-digit natural number such that the sum of its digits is 8}

Write the following set in roster form:

D = {x : x is a prime number which is divisor of 60}

Write the following set in roster form:

E = The set of all letters in the word TRIGONOMETRY

Write the following set in roster form:

F = The set of all letters in the word BETTER

Write the following set in the set-builder form:

{3, 6, 9, 12}

Write the following set in the set-builder form:

{2, 4, 8, 16, 32}

Write the following set in the set-builder form:

{5, 25, 125, 625}

Write the following set in the set-builder form:

{2, 4, 6, …}

Write the following set in the set-builder form:

{1, 4, 9, ....., 100}

List all the elements of the following set:

A = {x : x is an odd natural number}

List all the elements of the following set:

B = `{x : x  "is an integer", -1/2 < x < 9/2}`

List all the elements of the following set:

C = {x : x is an integer, x² ≤ 4}

List all the elements of the following set:

D = {x : x is a letter in the word “LOYAL”}

List all the elements of the following set:

E = {x : x is a month of a year not having 31 days}

List all the elements of the following set:

F = {x : x is a consonant in the English alphabet which precedes k}.

Match each of the set on the left in the roster form with the same set on the right described in set-builder form:

The following example is the null set example or not?

Set of odd natural numbers divisible by 2

The following example is the null set example or not?

Set of even prime numbers

The following example is the null set example or not?

{x : x is a natural numbers, x < 5 and x > 7}

The following example is the null set example or not?

{y : y is a point common to any two parallel lines}

Identify whether the following set is finite or infinite.

The set of months of a year

Identify whether the following set is finite or infinite.

{1, 2, 3, ...}

Identify whether the following set is finite or infinite.

{1, 2, 3, ... 99, 100}

Identify whether the following set is finite or infinite.

The set of positive integers greater than 100.

Identify whether the following set is finite or infinite.

The set of prime numbers less than 99.

State whether the following set is finite or infinite:

The set of lines which are parallel to the x-axis.

State whether the following set is finite or infinite:

The set of letters in the English alphabet.

State whether the following set is finite or infinite:

The set of numbers which are multiple of 5.

State whether the following set is finite or infinite:

The set of animals living on the earth.

State whether the following set is finite or infinite:

The set of circles passing through the origin (0, 0).

In the following, state whether A = B or not:

A = {a, b, c, d}; B = {d, c, b, a}

In the following, state whether A = B or not:

A = {4, 8, 12, 16}; B = {8, 4, 16, 18}

In the following, state whether A = B or not:

A = {2, 4, 6, 8, 10}; B = {x : x is positive even integer and x ≤ 10}

In the following, state whether A = B or not:

A = {x : x is a multiple of 10}; B = {10, 15, 20, 25, 30, ...}

Are the following pair of sets equal? Give reasons.

A = {2, 3}, B = {x : x is solution of x² + 5x + 6 = 0}

Are the following pair of sets equal? Give reasons.

A = {x : x is a letter in the word FOLLOW};

B = {y : y is a letter in the word WOLF}

From the sets given below, select equal sets:

A = {2, 4, 8, 12},
B = {1, 2, 3, 4},
C = {4, 8, 12, 14},
D = {3, 1, 4, 2},
E = {-1, 1},
F = {0, a},
G = {1, -1},
H = {0, 1}

Make correct statement by filling in the symbols ⊂ or ⊄ in the blank space:

{2, 3, 4} _____ {1, 2, 3, 4, 5}

Make correct statement by filling in the symbols ⊂ or ⊄ in the blank space:

{a, b, c} _____ {b, c, d}

Make correct statement by filling in the symbols ⊂ or ⊄ in the blank space:

{x : x is a student of Class XI of your school} ____ {x : x student of your school}

Make correct statement by filling in the symbols ⊂ or ⊄ in the blank space:

{x : x is a circle in the plane} _____ {x : x is a circle in the same plane with radius 1 unit}

Make correct statement by filling in the symbols ⊂ or ⊄ in the blank space:

{x : x is a triangle in a plane} _____ {x : x is a rectangle in the plane}

Make correct statement by filling in the symbols ⊂ or ⊄ in the blank space:

{x : x is an equilateral triangle in a plane} _____ {x : x is a triangle in the same plane}

Make correct statement by filling in the symbols ⊂ or ⊄ in the blank space:

{x : x is an even natural number} _____ {x : x is an integer}

{a, b} ⊄ {b, c, a}

{a, e} ⊂ {x : x is a vowel in the English alphabet}

{1, 2, 3} ⊂ {1, 3, 5}

{a} ⊂ {a. b, c}

{a} ∈ (a, b, c)

{x : x is an even natural number less than 6} ⊂ {x : x is a natural number which divides 36}

Let A = {1, 2, {3, 4}, 5}. The following statement is correct or incorrect and why?

{3, 4} ⊂ A

Let A = {1, 2, {3, 4}, 5}. The following statement is correct or incorrect and why?

{3, 4} ∈ A

Let A = {1, 2, {3, 4}, 5}. The following statement is correct or incorrect and why?

{{3, 4}} ⊂ A

Let A = {1, 2, {3, 4}, 5}. The following statement is correct or incorrect and why?

1 ∈ A

Let A = { 1, 2, { 3, 4}, 5 }. The following statement is correct or incorrect and why?

1 ⊂ A

Let A = {1, 2, {3, 4}, 5}. The following statement is correct or incorrect and why?

{1, 2, 5} ⊂ A

Let A = {1, 2, {3, 4}, 5}. The following statement is correct or incorrect and why?

{1, 2, 5} ∈ A

Let A = {1, 2, {3, 4}, 5}. The following statement is correct or incorrect and why?

{1, 2, 3} ⊂ A

Let A = {1, 2, {3, 4}, 5}. The following statement is correct or incorrect and why?

Φ ∈ A

Let A = {1, 2, {3, 4}, 5}. The following statement is correct or incorrect and why?

Φ ⊂ A

Let A = {1, 2, {3, 4}, 5}. The following statement is correct or incorrect and why?

{Φ} ⊂ A

Write down all the subsets of the following set:

{a}

Write down all the subsets of the following set:

{a, b}

Write down all the subsets of the following set:

{1, 2, 3}

Write down all the subsets of the following set:

Φ

Write the following as interval:

{x : x ∈ R, – 4 < x ≤ 6}

Write the following as intervals:  {x: x ∈ R, –12 < x < –10}

Write the following as intervals: {x : x ∈ R, 0 ≤ x < 7}

Write the following as intervals: {x : x ∈ R, 3 ≤ x ≤ 4}

Write the following interval in Set-Builder form:

(– 3, 0)

Write the following interval in Set-Builder form:

[6, 12]

Write the following interval in set-builder form:

(6, 12]

Write the following interval in set-builder form:

[–23, 5)

What universal set (s) would you propose for the following:

The set of right triangles.

Given the sets, A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, the following may be considered as universal set (s) for all the three sets A, B and C?

{0, 1, 2, 3, 4, 5, 6}

Given the sets, A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, the following may be considered as universal set (s) for all the three sets A, B and C?

Φ

Given the sets, A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, the following may be considered as universal set (s) for all the three sets A, B and C?

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, the following may be considered as universal set (s) for all the three sets A, B and C?

{1, 2, 3, 4, 5, 6, 7, 8}

Find the union of the following pairs of sets:

X = {1, 3, 5} Y = {1, 2, 3}

Find the union of the following pairs of sets:

A = {a, e, i, o, u}, B = {a, b, c} 

Find the union of the following pairs of sets:

A = {x : x is a natural number and multiple of 3}

B = {x : x is a natural number less than 6}

Find the union of the following pairs of sets:

A = {x : x is a natural number and 1 < x ≤ 6}

B = {x : x is a natural number and 6 < x < 10}

Find the union of the following pairs of sets:

A = {1, 2, 3}, B = Φ

Let A = {a, b}, B = {a, b, c}. Is A ⊂ B? What is A ∪ B?

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

If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find

A ∪ B

If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find

A ∪ C

If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find

B ∪ C

If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find

B ∪ D

If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find

A ∪ B ∪ C

If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find

A ∪ B ∪ D

If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find

B ∪ C ∪ D

Find the intersection of pair of sets:

X = {1, 3, 5}, Y = {1, 2, 3}

Find the intersection of pair of sets:

A = {a, e, i, o, u}, B = {a, b, c}

Find the intersection of pair of sets:

A = {x : x is a natural number and multiple of 3}

B = {x : x is a natural number less than 6}

Find the intersection of pair of sets:

A = {x : x is a natural number and 1 < x ≤ 6}

B = {x : x is a natural number and 6 < x < 10}

Find the intersection of pair of sets:

A = {1, 2, 3}, B = Φ

If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find

A ∩ B

If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find

B ∩ C

If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find

A ∩ C ∩ D

If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find

A ∩ C

If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find

B ∩ D

If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find

A ∩ (B ∪ C)

If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find

A ∩ D

If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find

A ∩ (B ∪ D)

If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find

(A ∩ B) ∩ (B ∪ C)

If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find

(A ∪ D) ∩ (B ∪ C)

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

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 ∩ C

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 ∩ D

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:

B ∩ C

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:

B ∩ D

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:

C ∩ D

State whether the following pairs of sets are disjoint.

{1, 2, 3, 4} and {x : x is a natural number and 4 ≤ x ≤ 6}

State whether the following pairs of sets are disjoint.

{a, e, i, o, u} and {c, d, e, f}

State whether the following pairs of sets are disjoint.

{x : x is an even integer} and {x : x is an odd integer}

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 A – B

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 A - C

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 A – D

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 B – A

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 C – A

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 D – A

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 B – C

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 B – D

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 C – B

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 D – B

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 C – D

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 D – C

If X = {a, b, c, d} and Y = {f, b, d, g}, find 

X – Y

If X = {a, b, c, d} and Y = {f, b, d, g}, find 

Y – X

If X = {a, b, c, d} and Y = {f, b, d, g}, find 

X ∩ Y

State whether the following statement is true or false. Justify your answer.

{2, 3, 4, 5} and {3, 6} are disjoint sets.

State whether the following statement is true or false. Justify your answer.

{a, e, i, o, u } and {a, b, c, d} are disjoint sets.

State whether the following statement is true or false. Justify your answer.

{2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.

State whether the following statement is true or false. Justify your answer.

{2, 6, 10} and {3, 7, 11} are disjoint sets.

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Find A'

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Find B'

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Find (A ∪ C)'

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Find (A ∪ B)'

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Find (A')'

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Find (B -C)'

If U = {a, b, c, d, e, f, g, h}, find the complements of the following set:

A = {a, b, c}

If U = {a, b, c, d, e, f, g, h}, find the complements of the following set:

B = {d, e, f, g}

If U = {a, b, c, d, e, f, g, h}, find the complements of the following set:

C = {a, c, e, g}

If U = {a, b, c, d, e, f, g, h}, find the complements of the following set:

D = {f, g, h, a}

Taking the set of natural numbers as the universal set, write down the complements of the following set:

{x : x is an even natural number}

Taking the set of natural numbers as the universal set, write down the complements of the following set:

{x : x is an odd natural number}

Taking the set of natural numbers as the universal set, write down the complements of the following set:

{x : x is a positive multiple of 3}

Taking the set of natural numbers as the universal set, write down the complements of the following set:

{x : x is a prime number}

Taking the set of natural numbers as the universal set, write down the complements of the following set:

{x : x is a natural number divisible by 3 and 5}

Taking the set of natural numbers as the universal set, write down the complements of the following set:

{x : x is a perfect square}

Taking the set of natural numbers as the universal set, write down the complements of the following set:

{x : x is perfect cube}

Taking the set of natural numbers as the universal set, write down the complements of the following set:

{x : x + 5 = 8}

Taking the set of natural numbers as the universal set, write down the complements of the following set:

{x : 2x + 5 = 9}

Taking the set of natural numbers as the universal set, write down the complements of the following set:

{x : x ≥ 7}

Taking the set of natural numbers as the universal set, write down the complements of the following set:

{x : x ∈ N and 2x + 1 > 10}

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that (A ∪ B)' = A' ∩ B'

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that (A ∩ B)' = A' ∪ B'

Draw appropriate Venn diagram for the following:

(A ∪ B)'

Draw appropriate Venn diagram for the following:

A' ∩ B'

Draw appropriate Venn diagram for the following:

(A ∩ B)'

Draw appropriate Venn diagram for the following:

A' ∪ B'

Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is A'?

Fill in the blank to make the following a true statement:

A ∪ A' = _____

Fill in the blank to make the following a true statement:

Φ′ ∩ A = _____

Fill in the blank to make the following a true statement:

A ∩ A' = _____

Fill in the blank to make the following a true statement:

U' ∩ A = _____

Decide, among the following sets, which sets are subsets of one and another:

A = {x : x ∈ R and x satisfy x² – 8x + 12 = 0},

B = {2, 4, 6}, C = {2, 4, 6, 8, …}, D = {6}.

Determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If x ∈ A and A ∈ B, then x ∈ B

Determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If A ⊂ B and B ∈ C, then A ∈ C

Determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If A ⊂ B and B ⊂ C, then A ⊂ C

Determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If A ⊄ B and B ⊄ C, then A ⊄ C

Determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If x ∈ A and A ⊄ B, then x ∈ B

Determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If A ⊂ B and x ∉ B, then x ∉ A

Let A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. Show that B = C.

Show that the following four conditions are equivalent:

1. A ⊂ B
2. A – B = Φ
3. A ∪ B = B 
4. A ∩ B = A

Show that if A ⊂ B, then C – B ⊂ C – A.

Show that for any sets A and B, A = (A ∩ B) ∪ ( A - B)

Show that for any sets A and B, A ∪ (B – A) = (A ∪ B)

Using properties of sets show that A ∪ (A ∩ B) = A

Using properties of sets show that A ∩ (A ∪ B) = A.

Show that A ∩ B = A ∩ C need not imply B = C.

Let A and B be sets. If A ∩ X = B ∩ X = Φ and A ∪ X = B ∪ X for some set X, show that A = B.

(Hints A = A ∩ (A ∪ X), B = B ∩ (B ∪ X) and use distributive law)

Find sets A, B and C such that A ∩ B, B ∩ C and A ∩ C are non-empty sets and A ∩ B ∩ C = Φ.