CHAPTER
2 Set Theory
INTRODUCTION REPRESENTATION OF SET
Fundamental concepts of set theory is a rich and beautiful. It There are two methods for representing a set.
permeates virtually every branch of mathematics. Yet, most (i) Tabulation method or Roster form
mathematics students receive only a cursory overview of the
All the elements belonging to the set are written in curly
theory of sets. brackets and separated by commas
A set is a collection of objects. The objects in such a collection If A is the set of days of a week, then
are called the elements of the set. We write a ∈ A to assert that
A = {Monday, Tuesday, Wednesday, Thursday, Friday,
a is an element, or a member, of the set A. We write a a ∉ A Saturday, Sunday}
when a is not an element of the set A. A set is merely the result of
(ii) Set Builder Method or Set rule method
collecting objects of interest and it is identified by enclosing its
elements with braces (curly brackets). For example the collection
In this method, we use the definition, which is satisfied by
all the elements of set.
A = {3, 7, 11, π} is a set that contains the four elements 3, 7, 11, π.
So 7 ∈ A, and 8 ∉ A. In above example set A may be written as
A = {x : x is a day of week}
DEFINITION NOTATIONS OF SET OF DIFFERENT
A set is a well defined collection of distinct objects. NUMBERS
(i)Set of all natural numbers N = {1, 2, 3, …}
(ii)Set of all integers Z or I = {0, ±1, ±2, …}
(iii)Set of non zero integers Z0 or I0 = {±1, ±2, ±3, …}
(iv) Set of all rational numbers
Q = {x : x = p/q, where p and q relatively prime integers and
Set of Fruits in a Set of Coin in a Set of Books in a q ≠ 0}
Basket Bank Library
Q0 denotes the set of all non−zero rational numbers.
Some examples of Set or not a Set (v) Set of real numbers is denoted by R
(vi) Set of complex numbers is denoted by C
1. Consonant in English Alphabet Set
2. Difficult topics in Mathematics Not a Set TYPES OF SETS
3. Collection of past Presidents of India Set (i) Null set or Void set or Empty set:
4. Group of Intelligent Students in JEE Batch Not a Set A set having no element is called as null set or empty set or
void set. It is denoted by φ or { }. The null set is unique and
Note: Generally, If you can see an adjective like good, difficult is the subset of every set.
intelligent, brave in a sentence then it does not describe a set. Examples
Sets are usually denoted by capital letters A, B, …, X, Y, Z. Set of even prime numbers less than 2.
Collection of objects or things in a set called as elements Set of natural numbers strictly lying between 5 and 6.
The elements of the set are denoted by small letters. A = {x : x ∈ N, 5 < x < 6} = f
,(ii) Finite Set
A set having contain either no elements or countable elements 2. The set {x : x is a positive integer less than 6 and 3x – 1
Example is an even number} in roster form is
Set of all natural numbers less than 6 = {1, 2, 3, 4, 5} (a) {1, 2, 3, 4, 5} (b) {1, 2, 3, 4, 5, 6}
(iii) Disjoint sets (c) {2, 4, 6} (d) {1, 3, 5}
Two sets A and B are said to be disjoint, if A ∩ B = φ. 3. Which of the following sets is a finite set?
If A ∩ B ≠ φ, then A and B are said to be intersecting or (a) A = {x ; x ∈ Z and x2 – 5x + 6 = 0}
overlapping sets. (b) B = {x ; x ∈ Z and x2 is even}
(iv) Singleton Set (c) D = {x ; x ∈ Z and x > –10}
A set having one and only one element is called singleton set (d) All of these
or unit set.
Example
Set of all positive integral roots of the equation
x2 – 2x – 15 = 0. CARDINAL NUMBER
⇒ (x + 3)(x – 5) = 0 The no. of distinct elements in a set A is denoted by n(A) and it is
⇒ x = –3 or x = 5 i.e., only one positive integral root. known as cardinal number of the set A.
C = {x ⇒ R : x – 5 = 0} Two finite sets A and B are equivalent if their cardinal number
⇒ x = 5 ⇒ C = {5} i.e., only one element. are same.
Example
A = {x ∈ Z and x2 – 5x + 6 = 0} ⇒ A = {2, 3}
Train Your Brain ∴ n(A) = 2
Example 1: Solve 3x2 − 12x = 0. EQUAL AND EQUIVALENT SETS
When Two finite set A and B are said to be Equivalent sets if their
(i) x ∈ N cardinalities are same i.e., n(A) = n(B).
(ii) x ∈ Z Two sets A and B are said to be Equal sets if not just their
(iii) x ∈ S where S = {a + ib : b ≠ 0; a, b ∈ R} cardinalities are same, but also the members in both the sets
are the same.
Sol. 3x2 − 12x = 0 ⇔ 3x (x − 4) = 0 ⇔ x = 0 or x = 4
If A and B are equal sets, we denote it by A = B.
(i) when x ∈N ⇒ x = 4
If A and B are unequal sets, we denote it by A ≠ B.
(ii) when x ∈ Z ⇒ x = 0 or x = 4
(iii) when x ∈ S ⇒ No solution
SUBSET AND SUPERSET
1
Example 2: = If Q = x:x , where y ∈ N , then
y If A and B are two sets such that every element of A is also an
element of B, then A is a subset of B and B is superset of A. We
(a) 0 ∈ Q (b) 1 ∈ Q
write A ⊆ B.
2
(c) 2 ∈ Q (d) ∈Q Note:
3
Every set is subset of itself i.e. A ⊆ A for all A.
Sol. (b) Since 1 ≠ 0, 1 ≠ 2, 1 ≠ −2 , [ y ∈ N ]
Empty set f is subset of every set
y y y 3
1 If A ⊆ B and B ⊆ A then A and B said to be equal sets, i.e.
∴ can be 1, [ y can be 1]. A = B.
y
If A ⊆ B and A ≠ B, there A is called as Proper subset of B and
denoted by A ⊂ B.
If a set A has n elements, then the number of subsets of A = 2n.
Concept Application POWER SET
1. A = {x : x ≠ x} represents Power set of a set A is the collection of all subsets of A and is
(a) {0} (b) { } denoted by P(A) or 2n.
(c) {1} (d) {x} Let A be a finite set containing m elements i.e., n(A) = m, then
30 JEE (XI) Module-1 PW
, The number of elements in the power set of A,
n(P(A)) = 2m.
The number of non-void/non-empty subsets of
A = (2m) –1.
Concept Application
The number of proper subsets of A = 2m – 1. 4. Consider the following sets.
The number of non-void proper subsets of A = 2m – 2. I. A = {1, 2, 3}
Example: II. B = {x ∈ R : x2 – 2x + 1 = 0}
The number of elements in the power set of set A = {1, 2} III. C = {1, 2, 2, 3}
is 22. IV. D = {x ∈ R : x3 – 6x2 + 11x – 6 = 0}
Which of the following are equal?
UNIVERSAL SET (a) A = B = C (b) A = C = D
(c) A = B = D (d) B = C = D
The universal set is the superset for all the sets under the
consideration. 5. The number of the proper subset of {a, b, c} is:
The set of complex numbers is the universal set for all possible (a) 3 (b) 8 (c) 6 (d) 7
sets related numbers. 6. Given the sets
Note: A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}.
In a set, the order in which elements are written makes no Which of the following may be considered as universal
difference. set for all the three sets A, B and C?
In a set, the repetition of elements has no etc. (a) {0, 1, 2, 3, 4, 5, 6}
(b) f
(c) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Train Your Brain (d) {1, 2, 3, 4, 5, 6, 7, 8}
7. The cardinality of the set P{P[P(f)]} is
Example 3: The number of non-empty subsets of the set (a) 0 (b) 1 (c) 2 (d) 4
{1, 2, 3, 4} is
(a) 15 (b) 14 (c) 16 (d) 17
Sol. (a) The number of non- empty subsets INTERVALS AS SUBSETS OF R
= 2n – 1 = 24 – 1 = 16 – 1 = 15.
Four type of subsets can be defined on R as given below.
Example 4: Consider the following sets.
A=0 Let a, b ∈ R, such that a < b
B = {x : x > 15 and x < 5}, 1. Open Interval
C = {x : x – 5 = 0}, (a, b) or] a, b [= {x : a < x < b}
D = {x : x2 = 25}, = Set of all real numbers between a and b, not including
E = {x : x is an integral positive root of the equation a and b both.
x2 – 2x – 15 = 0}
Choose the pair of equal sets a b
(a) A and B (b) C and D ]a, b[ or (a, b)
(c) C and E (d) B and C
2. Closed Interval
Sol. (c) 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, b] = {x : a ≤ x ≤ b}
A ≠ D, A ≠ E. Since, B = f, but none of the other = Set of all real numbers between a and b as well as including
sets are empty. Therefore B ≠ C, B ≠ D and B ≠ E. a and b both.
Also, C = {5} but – 5 ∈ D, hence C ≠ D.
Since, E = {5}, C = E. Further, D = {–5, 5} and sets a b
is C and E. [a, b]
Example 5: If A = {x, y} then the power set of A is 3. Open-closed Interval (semi closed or semi open interval)
(a) {xy, yx} (b) {f, x, y} (a, b] or ]a, b] = {x : a < x ≤ b}
(c) {f, {x}, {2y}} (d) {f, {x} {y}, {x, y}} = Set of all real numbers between a and b, a not included but
Sol. (d) The collection of all the subsets of the set A is b included.
called the power set of A. It is denoted by P(A)
Given A = {x, y}; P(A) = {f,{x}, {y}, {x, y}} a b
(a, b] or ] a, b]
P
W Sets 31
2 Set Theory
INTRODUCTION REPRESENTATION OF SET
Fundamental concepts of set theory is a rich and beautiful. It There are two methods for representing a set.
permeates virtually every branch of mathematics. Yet, most (i) Tabulation method or Roster form
mathematics students receive only a cursory overview of the
All the elements belonging to the set are written in curly
theory of sets. brackets and separated by commas
A set is a collection of objects. The objects in such a collection If A is the set of days of a week, then
are called the elements of the set. We write a ∈ A to assert that
A = {Monday, Tuesday, Wednesday, Thursday, Friday,
a is an element, or a member, of the set A. We write a a ∉ A Saturday, Sunday}
when a is not an element of the set A. A set is merely the result of
(ii) Set Builder Method or Set rule method
collecting objects of interest and it is identified by enclosing its
elements with braces (curly brackets). For example the collection
In this method, we use the definition, which is satisfied by
all the elements of set.
A = {3, 7, 11, π} is a set that contains the four elements 3, 7, 11, π.
So 7 ∈ A, and 8 ∉ A. In above example set A may be written as
A = {x : x is a day of week}
DEFINITION NOTATIONS OF SET OF DIFFERENT
A set is a well defined collection of distinct objects. NUMBERS
(i)Set of all natural numbers N = {1, 2, 3, …}
(ii)Set of all integers Z or I = {0, ±1, ±2, …}
(iii)Set of non zero integers Z0 or I0 = {±1, ±2, ±3, …}
(iv) Set of all rational numbers
Q = {x : x = p/q, where p and q relatively prime integers and
Set of Fruits in a Set of Coin in a Set of Books in a q ≠ 0}
Basket Bank Library
Q0 denotes the set of all non−zero rational numbers.
Some examples of Set or not a Set (v) Set of real numbers is denoted by R
(vi) Set of complex numbers is denoted by C
1. Consonant in English Alphabet Set
2. Difficult topics in Mathematics Not a Set TYPES OF SETS
3. Collection of past Presidents of India Set (i) Null set or Void set or Empty set:
4. Group of Intelligent Students in JEE Batch Not a Set A set having no element is called as null set or empty set or
void set. It is denoted by φ or { }. The null set is unique and
Note: Generally, If you can see an adjective like good, difficult is the subset of every set.
intelligent, brave in a sentence then it does not describe a set. Examples
Sets are usually denoted by capital letters A, B, …, X, Y, Z. Set of even prime numbers less than 2.
Collection of objects or things in a set called as elements Set of natural numbers strictly lying between 5 and 6.
The elements of the set are denoted by small letters. A = {x : x ∈ N, 5 < x < 6} = f
,(ii) Finite Set
A set having contain either no elements or countable elements 2. The set {x : x is a positive integer less than 6 and 3x – 1
Example is an even number} in roster form is
Set of all natural numbers less than 6 = {1, 2, 3, 4, 5} (a) {1, 2, 3, 4, 5} (b) {1, 2, 3, 4, 5, 6}
(iii) Disjoint sets (c) {2, 4, 6} (d) {1, 3, 5}
Two sets A and B are said to be disjoint, if A ∩ B = φ. 3. Which of the following sets is a finite set?
If A ∩ B ≠ φ, then A and B are said to be intersecting or (a) A = {x ; x ∈ Z and x2 – 5x + 6 = 0}
overlapping sets. (b) B = {x ; x ∈ Z and x2 is even}
(iv) Singleton Set (c) D = {x ; x ∈ Z and x > –10}
A set having one and only one element is called singleton set (d) All of these
or unit set.
Example
Set of all positive integral roots of the equation
x2 – 2x – 15 = 0. CARDINAL NUMBER
⇒ (x + 3)(x – 5) = 0 The no. of distinct elements in a set A is denoted by n(A) and it is
⇒ x = –3 or x = 5 i.e., only one positive integral root. known as cardinal number of the set A.
C = {x ⇒ R : x – 5 = 0} Two finite sets A and B are equivalent if their cardinal number
⇒ x = 5 ⇒ C = {5} i.e., only one element. are same.
Example
A = {x ∈ Z and x2 – 5x + 6 = 0} ⇒ A = {2, 3}
Train Your Brain ∴ n(A) = 2
Example 1: Solve 3x2 − 12x = 0. EQUAL AND EQUIVALENT SETS
When Two finite set A and B are said to be Equivalent sets if their
(i) x ∈ N cardinalities are same i.e., n(A) = n(B).
(ii) x ∈ Z Two sets A and B are said to be Equal sets if not just their
(iii) x ∈ S where S = {a + ib : b ≠ 0; a, b ∈ R} cardinalities are same, but also the members in both the sets
are the same.
Sol. 3x2 − 12x = 0 ⇔ 3x (x − 4) = 0 ⇔ x = 0 or x = 4
If A and B are equal sets, we denote it by A = B.
(i) when x ∈N ⇒ x = 4
If A and B are unequal sets, we denote it by A ≠ B.
(ii) when x ∈ Z ⇒ x = 0 or x = 4
(iii) when x ∈ S ⇒ No solution
SUBSET AND SUPERSET
1
Example 2: = If Q = x:x , where y ∈ N , then
y If A and B are two sets such that every element of A is also an
element of B, then A is a subset of B and B is superset of A. We
(a) 0 ∈ Q (b) 1 ∈ Q
write A ⊆ B.
2
(c) 2 ∈ Q (d) ∈Q Note:
3
Every set is subset of itself i.e. A ⊆ A for all A.
Sol. (b) Since 1 ≠ 0, 1 ≠ 2, 1 ≠ −2 , [ y ∈ N ]
Empty set f is subset of every set
y y y 3
1 If A ⊆ B and B ⊆ A then A and B said to be equal sets, i.e.
∴ can be 1, [ y can be 1]. A = B.
y
If A ⊆ B and A ≠ B, there A is called as Proper subset of B and
denoted by A ⊂ B.
If a set A has n elements, then the number of subsets of A = 2n.
Concept Application POWER SET
1. A = {x : x ≠ x} represents Power set of a set A is the collection of all subsets of A and is
(a) {0} (b) { } denoted by P(A) or 2n.
(c) {1} (d) {x} Let A be a finite set containing m elements i.e., n(A) = m, then
30 JEE (XI) Module-1 PW
, The number of elements in the power set of A,
n(P(A)) = 2m.
The number of non-void/non-empty subsets of
A = (2m) –1.
Concept Application
The number of proper subsets of A = 2m – 1. 4. Consider the following sets.
The number of non-void proper subsets of A = 2m – 2. I. A = {1, 2, 3}
Example: II. B = {x ∈ R : x2 – 2x + 1 = 0}
The number of elements in the power set of set A = {1, 2} III. C = {1, 2, 2, 3}
is 22. IV. D = {x ∈ R : x3 – 6x2 + 11x – 6 = 0}
Which of the following are equal?
UNIVERSAL SET (a) A = B = C (b) A = C = D
(c) A = B = D (d) B = C = D
The universal set is the superset for all the sets under the
consideration. 5. The number of the proper subset of {a, b, c} is:
The set of complex numbers is the universal set for all possible (a) 3 (b) 8 (c) 6 (d) 7
sets related numbers. 6. Given the sets
Note: A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}.
In a set, the order in which elements are written makes no Which of the following may be considered as universal
difference. set for all the three sets A, B and C?
In a set, the repetition of elements has no etc. (a) {0, 1, 2, 3, 4, 5, 6}
(b) f
(c) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Train Your Brain (d) {1, 2, 3, 4, 5, 6, 7, 8}
7. The cardinality of the set P{P[P(f)]} is
Example 3: The number of non-empty subsets of the set (a) 0 (b) 1 (c) 2 (d) 4
{1, 2, 3, 4} is
(a) 15 (b) 14 (c) 16 (d) 17
Sol. (a) The number of non- empty subsets INTERVALS AS SUBSETS OF R
= 2n – 1 = 24 – 1 = 16 – 1 = 15.
Four type of subsets can be defined on R as given below.
Example 4: Consider the following sets.
A=0 Let a, b ∈ R, such that a < b
B = {x : x > 15 and x < 5}, 1. Open Interval
C = {x : x – 5 = 0}, (a, b) or] a, b [= {x : a < x < b}
D = {x : x2 = 25}, = Set of all real numbers between a and b, not including
E = {x : x is an integral positive root of the equation a and b both.
x2 – 2x – 15 = 0}
Choose the pair of equal sets a b
(a) A and B (b) C and D ]a, b[ or (a, b)
(c) C and E (d) B and C
2. Closed Interval
Sol. (c) 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, b] = {x : a ≤ x ≤ b}
A ≠ D, A ≠ E. Since, B = f, but none of the other = Set of all real numbers between a and b as well as including
sets are empty. Therefore B ≠ C, B ≠ D and B ≠ E. a and b both.
Also, C = {5} but – 5 ∈ D, hence C ≠ D.
Since, E = {5}, C = E. Further, D = {–5, 5} and sets a b
is C and E. [a, b]
Example 5: If A = {x, y} then the power set of A is 3. Open-closed Interval (semi closed or semi open interval)
(a) {xy, yx} (b) {f, x, y} (a, b] or ]a, b] = {x : a < x ≤ b}
(c) {f, {x}, {2y}} (d) {f, {x} {y}, {x, y}} = Set of all real numbers between a and b, a not included but
Sol. (d) The collection of all the subsets of the set A is b included.
called the power set of A. It is denoted by P(A)
Given A = {x, y}; P(A) = {f,{x}, {y}, {x, y}} a b
(a, b] or ] a, b]
P
W Sets 31
