NCERT In Short

Class:-11(sci.) Sub:-MATHS
Ch:-1 SETS

Example 1 Write the solution set of the equation x 2 + x – 2=0 in roster form.
Solution The given equation can be written as
( x – 1)( x+2)=0 ,i .e . , x=1 , – 2
Therefore, the solution set of the given equation can be written in roster form as {1, –2}.

Example 2 Write the set {x : x is a positive integer and x2 < 40} in the roster form.
Solution The required numbers are 1, 2, 3, 4, 5, 6. So, the given set in the roster form is {1, 2, 3, 4, 5,
6}.

Example 3 Write the set A = {1, 4, 9, 16, 25, . . . }in set-builder form.
Solution We may write the set A as
A = {x : x is the square of a natural number}
Alternatively, we can write
A = {x : x = n2, where n ∈N}

Example 4 Write the set [

1 2 3 4 5 6
, , , , ,
2 3 4 5 6 7 ]
in the set-builder form.
Solution We see that each member in the given set has the numerator one less than the denominator.
Also, the numerator begin from 1 and do not exceed 6. Hence, in the set-builder form the given set is
n
{x:x= , n where is a natural number and 1≤ n≤ 6 }
(n+1)

Example 5 Match each of the set on the left described in the roster form with the same set on the right
described in the set-builder form :
(i) {P, R, I, N, C, A, L} (a) { x : x is a positive integer and is a divisor of 18}
(ii) { 0 } (b) { x : x is an integer and x2 – 9 = 0 }
(iii) {1, 2, 3, 6, 9, 18} (c) {x : x is an integer and x + 1= 1}
(iv) {3, –3} (d) {x : x is a letter of the word PRINCIPAL}
Solution Since in (d), there are 9 letters in the word PRINCIPAL and two letters P and
are repeated, so (i) matches (d). Similarly, (ii) matches (c) as x + 1 = 1 implies x = 0. Also, 1, 2 ,3, 6, 9,
18 are all divisors of 18 and so (iii) matches (a). Finally, x2 – 9 = 0 implies x = 3, –3 and so (iv)
matches (b).

EXERCISE 1.1

1. Which of the following are sets ? Justify your answer.

(i) The collection of all the months of a year beginning with the 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 natural numbers less than 100.
(vi) A collection of novels written by the writer Munshi Prem Chand.
(vii) The collection of all even integers.
(viii) The collection of questions in this Chapter.
(ix) A collection of most dangerous animals of the world
2. Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈or ∉in the blank spaces:
(i) 5. . .A (ii) 8. . .A (iii) 0. . .A (iv) 4. . . A (v) 2. . .A (vi) 10. . .A
3. Write the following sets in roster form:
(i) A = {x : x is an integer and –3 ≤ x < 7}
(ii) B = {x : x is a natural number less than 6}
(iii) C = {x : x is a two-digit natural number such that the sum of its digits is 8}
(iv) D = {x : x is a prime number which is divisor of 60}
(v) E = The set of all letters in the word TRIGONOMETRY
(vi) F = The set of all letters in the word BETTER
4. 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) {2, 4, 6, . . .} (v) {1,4,9, . . .,100}
5. List all the elements of the following sets :
(i) A = {x : x is an odd natural number}
1 9
(ii) B = {x : x is an integer, − < x < }
2 2
(iii) C = {x : x is an integer, x2 ≤ 4}
(iv) D = {x : x is a letter in the word “LOYAL”}
(v) E = {x : x is a month of a year not having 31 days}
(vi) F = {x : x is a consonant in the English alphabet which precedes k }.
6. Match each of the set on the left in the roster form with the same set on the right described in 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 less than 10}
(iii){M,A,T,H,E,I,C,S} (c) {x : x is natural number and divisor of 6}
(iv) {1, 3, 5, 7, 9} (d) {x : x is a letter of the word MATHEMATICS}

Example 6 State which of the following sets are finite or infinite :

(i) {x : x ∈N and (x – 1) (x –2) = 0}
(ii) {x : x ∈N and x2 = 4}
(iii) {x : x ∈N and 2x –1 = 0}
(iv) {x : x ∈N and x is prime}
(v) {x : x ∈N and x is odd}
Solution (i) Given set = {1, 2}. Hence, it is finite.
(ii) Given set = {2}. Hence, it is finite.
(iii) Given set = φ. Hence, it is finite.
(iv) The given set is the set of all prime numbers and since set of prime numbers is
infinite. Hence the given set is infinite.
(v) Since there are infinite number of odd numbers, hence, the given set is infinite.
Example 7 Find the pairs of equal sets, if any, give reasons:
A = {0}, B = {x : x > 15 and x < 5}
C = {x : x – 5 = 0 }, D = {x: x2 = 25},
E = {x : x is an integral positive root of the equation x2 – 2x –15 = 0}.
Solution Since 0 ∈A and 0 does not belong to any of the sets B, C, D and E, it follows that, A ≠ B,
A ≠ C, A ≠ D, A ≠ E.
Since B = φ but none of the other sets are empty. Therefore B ≠ C, B ≠ D and B ≠ E. Also
C = {5} but –5 ∈D, hence C ≠ D.
Since E = {5}, C = E. Further, D = {–5, 5} and E = {5}, we find that, D ≠ E. Thus, the only pair
of equal sets is C and E.

Example 8 Which of the following pairs of sets are equal? Justify your answer.
(i) X, the set of letters in “ALLOY” and B, the set of letters in “LOYAL”.
(ii) A = {n : n ∈Z and n2 ≤ 4} and B = {x : x ∈R and x2 – 3x + 2 = 0}.
Solution (i) We have, X = {A, L, L, O, Y}, B = {L, O, Y, A, L}. Then X and B are equal sets as
repetition of elements in a set do not change a set. Thus,
X = {A, L, O, Y} = B
(ii) A = {–2, –1, 0, 1, 2}, B = {1, 2}. Since 0 ∈A and 0 ∉B, A and B are not equal sets.

EXERCISE 1.2

1. Which of the following are examples of the null set

(i) Set of odd natural numbers divisible by 2
(ii) Set of even prime numbers
(iii) { x : x is a natural numbers, x < 5 and x > 7 }
(iv) { y : y is a point common to any two parallel lines}
2. Which of the following sets are finite or infinite
(i) The set of months of a year
(ii) {1, 2, 3, . . .}
(iii) {1, 2, 3, . . .99, 100}
(iv) The set of positive integers greater than 100
(v) The set of prime numbers less than 99
3. State whether each of the following set is finite or infinite:
(i) The set of lines which are parallel to the x-axis
(ii) The set of letters in the English alphabet
(iii) The set of numbers which are multiple of 5
(iv) The set of animals living on the earth
(v) The set of circles passing through the origin (0,0)
4. In the following, state whether A = B or not:
(i) A = { a, b, c, d } B = { d, c, b, a }
(ii) A = { 4, 8, 12, 16 } B = { 8, 4, 16, 18}
(iii) A = {2, 4, 6, 8, 10} B = { x : x is positive even integer and x ≤ 10}
(iv) A = { x : x is a multiple of 10} B = { 10, 15, 20, 25, 30, . . . }
5. Are the following pair of sets equal ? Give reasons.
(i) A = {2, 3}, B = {x : x is solution of x2 + 5x + 6 = 0}
(ii) A = { x : x is a letter in the word FOLLOW}
B = { y : y is a letter in the word WOLF}
6. B = { y : y is a letter in the word WOLF}
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}

Example 9 Consider the sets

φ, A = { 1, 3 }, B = {1, 5, 9}, C = {1, 3, 5, 7, 9}.
Insert the symbol ⊂or ⊄between each of the following pair of sets:
(i) φ . . . B (ii) A . . . B (iii) A . . . C (iv) B . . . C
Solution (i) φ ⊂B as φ is a subset of every set.
(ii) A ⊄B as 3 ∈A and 3 ∉B
(iii) A ⊂C as 1, 3 ∈A also belongs to C
(iv) B ⊂C as each element of B is also an element of C.

Example 10 Let A = { a, e, i, o, u} and B = { a, b, c, d}. Is A a subset of B ? No. (Why?). Is B a subset

of A? No. (Why?)

Example 11 Let A, B and C be three sets. If A ∈B and B ⊂C, is it true that A ⊂C?. If not, give an
example.
Solution No. Let A = {1}, B = {{1}, 2} and C = {{1}, 2, 3}. Here A ∈B as A = {1} and B ⊂C. But A ⊄
C as 1 ∈A and 1 ∉C.
Note that an element of a set can never be a subset of itself

EXERCISE 1.3

1. Make correct statements by filling in the symbols ⊂or ⊄in the blank spaces :
(i) { 2, 3, 4 } . . . { 1, 2, 3, 4,5 } (ii) { a, b, c } . . . { b, c, d }
(ii) {x : x is a student of Class XI of your school}. . .{x : x student of your school}
(iv) {x : x is a circle in the plane} . . .{x : x is a circle in the same plane with radius 1 unit}
(v) {x : x is a triangle in a plane} . . . {x : x is a rectangle in the plane}
(vi) {x : x is an equilateral triangle in a plane} . . . {x : x is a triangle in the same plane}
(vii) {x : x is an even natural number} . . . {x : x is an integer}
2. Examine whether the following statements are true or false:
(i) { a, b } ⊄{ b, c, a }
(ii) { a, e } ⊂{ x : x is a vowel in the English alphabet}
(iii) { 1, 2, 3 } ⊂{ 1, 3, 5 }
(iv) { a }⊂ { a, b, c }
(v) { a }∈ { a, b, c }
(vi){ x : x is an even natural number less than 6} ⊂{x : x is a natural number which divides 36}
3. Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?
(i) {3, 4} ⊂A (ii) {3, 4} ∈A (iii) {{3, 4}} ⊂A
(iv) 1 ∈A (v) 1 ⊂A (vi) {1, 2, 5} ⊂A
(vii) {1, 2, 5} ∈A (viii){1, 2, 3} ⊂A (ix) φ ∈A
(x) φ ⊂A (xi) {φ} ⊂A
4. Write down all the subsets of the following sets
(i) {a} (ii) {a, b} (iii) {1, 2, 3} (iv) φ
5. Write the following as intervals :
(i) {x : x ∈R, – 4 < x ≤ 6} (ii) {x : x ∈R, – 12 < x < –10}
(iii) {x : x ∈R, 0 ≤ x < 7} (iv) {x : x ∈R, 3 ≤ x ≤ 4}
6. Write the following intervals in set-builder form :
(i) (– 3, 0) (ii) [6 , 12] (iii) (6, 12] (iv) [–23, 5)
7. What universal set(s) would you propose for each of the following :
(i) The set of right triangles. (ii) The set of isosceles triangles.
8. Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be
considered as universal set (s) for all the three sets A, B and C
(i) {0, 1, 2, 3, 4, 5, 6}
(ii) φ
(iii) {0,1,2,3,4,5,6,7,8,9,10}
(iv) {1,2,3,4,5,6,7,8}

Example 12 Let A = { 2, 4, 6, 8} and B = { 6, 8, 10, 12}. Find A ∪B.

Solution We have A ∪ B = { 2, 4, 6, 8, 10, 12}
Note that the common elements 6 and 8 have been taken only once while writing A ∪B.

Example 13 Let A = { a, e, i, o, u } and B = { a, i, u }. Show that A ∪B = A

Solution We have, A ∪ B = { a, e, i, o, u } = A.
This example illustrates that union of sets A and its subset B is the set A itself, i.e., if B ⊂A,
then A ∪B = A.

Example 14 Let X = {Ram, Geeta, Akbar} be the set of students of Class XI, who are in school hockey
team. Let Y = {Geeta, David, Ashok} be the set of students from Class XI who are in the school
football team. Find X ∪Y and interpret the set.
Solution We have, X ∪Y = {Ram, Geeta, Akbar, David, Ashok}. This is the set of students from Class
XI who are in the hockey team or the football team or both.
Thus, we can define the union of two sets as follows

Example 15 Consider the sets A and B of Example 12. Find A ∩ B.

Solution We see that 6, 8 are the only elements which are common to both A and B. Hence A ∩ B =
{ 6, 8 }.

Example 16 Consider the sets X and Y of Example 14. Find X ∩ Y.

Solution We see that element ‘Geeta’ is the only element common to both. Hence, X ∩ Y = {Geeta}.

Example 17 Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and B = { 2, 3, 5, 7 }. Find A ∩ B and hence show

that A ∩ B = B.
Solution We have A ∩ B = { 2, 3, 5, 7 } = B. We note that B ⊂A and that A ∩ B = B.

Example 18 Let A = { 1, 2, 3, 4, 5, 6}, B = { 2, 4, 6, 8 }. Find A – B and B – A.

Solution We have, A – B = { 1, 3, 5 }, since the elements 1, 3, 5 belong to A but not to B and B – A =
{ 8 }, since the element 8 belongs to B and not to A. We note that A – B ≠ B – A.

Example 19 Let V = { a, e, i, o, u } and B = { a, i, k, u}. Find V – B and B – V

Solution We have, V – B = { e, o }, since the elements e, o belong to V but not to B and B – V = { k },
since the element k belongs to B but not to V.
We note that V – B ≠ B – V. Using the set builder notation, we can rewrite the definition of
difference as
A – B = { x : x ∈A and x ∉ B }
The difference of two sets A and B can be represented by Venn diagram as shown in Fig 1.8.
The shaded portion represents the difference of the two sets A and B.

EXERCISE 1.4

1. Find the union of each of the following pairs of sets :

(i) X = {1, 3, 5} Y = {1, 2, 3}
(ii) A = [ a, e, i, o, u} B = {a, b, c}
(iii) A = {x : x is a natural number and multiple of 3}
B = {x : x is a natural number less than 6}
(iv) A = {x : x is a natural number and 1 < x ≤6 }
B = {x : x is a natural number and 6 < x < 10 }
(v) A = {1, 2, 3}, B = φ
2. Let A = { a, b }, B = {a, b, c}. Is A ⊂B ? What is A ∪B ?
3. If A and B are two sets such that A ⊂B, then what is A ∪B ?
4. If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find
(i) A ∪B (ii) A ∪C (iii) B ∪C (iv) B ∪D
(v) A ∪ B ∪C (vi) A ∪B ∪D (vii) B ∪C ∪D
5. Find the intersection of each pair of sets of question 1 above.
6. If A = { 3, 5, 7, 9, 11 }, B = {7, 9, 11, 13}, C = {11, 13, 15}and D = {15, 17}; find
(i) A ∩ B (ii) B ∩ C (iii) A ∩ C ∩ D
(iv) A ∩ C (v) B ∩ D (vi) A ∩ (B ∪C)
(vii) A ∩ D (viii) A ∩ (B ∪D) (ix) ( A ∩ B ) ∩ ( B ∪C )
(x) ( A ∪D) ∩ ( B ∪C)
7. If A = {x : x is a natural number }, B = {x : x is an even natural number} C = {x : x is an odd natural
number}andD = {x : x is a prime number }, find
(i) A ∩ B (ii) A ∩ C (iii) A ∩ D
(iv) B ∩ C (v) B ∩ D (vi) C ∩ D

8. Which of the following pairs of sets are disjoint

(i) {1, 2, 3, 4} and {x : x is a natural number and 4 ≤ x ≤ 6 }
(ii) { a, e, i, o, u } and { c, d, e, f }
(iii) {x : x is an even integer } and {x : x is an odd integer}
9. 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 (xi) C – D (xii) D – C
10. If X= { a, b, c, d } and Y = { f, b, d, g}, find
(i) X – Y (ii) Y – X (iii) X ∩ Y
11. If R is the set of real numbers and Q is the set of rational numbers, then what is R – Q?
12. State whether each of the following statement is true or false. Justify your answer.
 (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.

Example 20 Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 3, 5, 7, 9}. Find A′.

Solution We note that 2, 4, 6, 8, 10 are the only elements of U which do not belong to A. Hence
A′ = { 2, 4, 6, 8,10 }.

Example 21 Let U be universal set of all the students of Class XI of a coeducational school and A be
the set of all girls in Class XI. Find A′.
Solution Since A is the set of all girls, A′ is clearly the set of all boys in the class.

Example 22 Let U = {1, 2, 3, 4, 5, 6}, A = {2, 3} and B = {3, 4, 5}. Find A′, B′ , A′ ∩ B′, A ∪B and
hence show that ( A ∪B )′ = A′ ∩ B′.
Solution Clearly A′ = {1, 4, 5, 6}, B′ = { 1, 2, 6 }. Hence A′ ∩ B′ = { 1, 6 } Also A ∪B = { 2, 3, 4, 5 },
so that (A ∪B)′ ( A ∪B )′ = { 1, 6 }
( A ∪B )′ = { 1, 6 } = A′ ∩ B′

EXERCISE 1.5

1. 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

(i) A′ (ii) B′ (iii) (A ∪C)′ (iv) (A ∪B)′ (v) (A′)′
(vi) (B – C)′
2. If U = { a, b, c, d, e, f, g, h}, find the complements of the following sets :
(i) A = {a, b, c} (ii) B = {d, e, f, g}
(iii) C = {a, c, e, g} (iv) D = { f, g, h, a}
3. Taking the set of natural numbers as the universal set, write down the complements of the following
sets:
(i) {x : x is an even natural number} (ii) { x : x is an odd natural number }
(iii) {x : x is a positive multiple of 3} (iv) { x : x is a prime number }
(v) {x : x is a natural number divisible by 3 and 5}
(vi) { x : x is a perfect square } (vii) { x : x is a perfect cube}
(viii) { x : x + 5 = 8 } (ix) { x : 2x + 5 = 9}
(x) { x : x ≥ 7 } (xi) { x : x ∈N and 2x + 1 > 10 }
4. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = {2, 4, 6, 8} and B = { 2, 3, 5, 7}. Verify that
(i) (A ∪B)′ = A′ ∩ B′ (ii) (A ∩ B)′ = A′ ∪B′
5. Draw appropriate Venn diagram for each of the following :
(i) (A ∪B)′, (ii) A′ ∩ B′, (iii) (A ∩ B)′, (iv) A′ ∪B′
6. 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′?
7. Fill in the blanks to make each of the following a true statement :
(i) A ∪A′ = . . . (ii) φ′ ∩ A = . . .
(iii) A ∩ A′ = . . . (iv) U′ ∩ A = . . .
(iv)
 Miscellaneous Examples

Example 23 Show that the set of letters needed to spell “ CATARACT ” and the set of letters needed to
spell “ TRACT” are equal.
Solution Let X be the set of letters in “CATARACT”. Then
X = { C, A, T, R }
Let Y be the set of letters in “ TRACT”. Then
Y = { T, R, A, C, T } = { T, R, A, C }
Since every element in X is in Y and every element in Y is in X. It follows that X = Y.

Example 24 List all the subsets of the set { –1, 0, 1 }.

Solution Let A = { –1, 0, 1 }. The subset of A having no element is the empty set φ. The subsets of A
having one element are { –1 }, { 0 }, { 1 }. The subsets of A having two elements are {–1, 0}, {–1, 1} ,
{0, 1}. The subset of A having three elements of A is A itself. So, all the subsets of A are φ, {–1}, {0},
{1}, {–1, 0}, {–1, 1}, {0, 1} and {–1, 0, 1}.

Example 25 Show that A ∪B = A ∩ B implies A = B

Solution Let a ∈A. Then a ∈A ∪B. Since A ∪B = A ∩ B , a ∈A ∩ B. So a ∈B. Therefore, A ⊂B.
Similarly, if b ∈B, then b ∈A ∪B. Since
A ∪ B = A ∩ B, b ∈A ∩ B. So, b ∈A. Therefore, B ⊂A. Thus, A = B

Miscellaneous Exercise on Chapter 1

1. 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 }.
2. In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is
false, give an example.
(i) If x ∈A and A ∈B , then x ∈B
(ii) If A ⊂B and B ∈C , then A ∈C
(iii) If A ⊂B and B ⊂C , then A ⊂C
(iv) If A ⊄B and B ⊄C , then A ⊄C
(v) If x ∈A and A ⊄B , then x ∈B
(vi) If A ⊂B and x ∉B , then x ∉A
3. Let A, B, and C be the sets such that A ∪B = A ∪C and A ∩ B = A ∩ C. Show that B = C.
4. Show that the following four conditions are equivalent :
(i) A ⊂B (ii) A – B = φ (iii) A ∪B = B (iv) A ∩ B = A
5. Show that if A ⊂B, then C – B ⊂C – A.
6. Show that for any sets A and B,
A = ( A ∩ B ) ∪( A – B ) and A ∪( B – A ) = ( A ∪B )
7.Using properties of sets, show that
(i) A ∪( A ∩ B ) = A (ii) A ∩ ( A ∪B ) = A.
8. Show that A ∩ B = A ∩ C need not imply B = C.
9. 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 )
10. Find sets A, B and C such that A ∩ B, B ∩ C and A ∩ C are non-empty sets and A ∩ B ∩ C = φ.