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Class 11 | CBSE | MATHEMATICS
CHAPTER 1 | TYPES OF SETS

MATHEMATICS CLASS 11 :

1.3 THE EMPTY SET (Null / Void Set)


πŸ”Ή Introduction

Consider the set:

A = {x : x is a student of Class XI presently studying in a school}

The number of students in Class XI can be counted. Hence, set A contains a finite number of elements.

Now consider another set:

B = {x : x is a student presently studying in both Classes X and XI}

A student cannot study in two different classes at the same time. Hence, set B contains no element.

πŸ“Œ Definition 1

A set which does not contain any element is called the empty set, null set, or void set.

The empty set is denoted by:

\[ \varnothing \quad \text{or} \quad \{\} \]

Examples of Empty Sets

  1. A = {x : 1 < x < 2, x is a natural number}

    There is no natural number between 1 and 2.

    \[ A = \varnothing \]

  2. B = {x : xΒ² - 2 = 0, x is a rational number}

    \[ x^2 = 2 \] \[ x = \sqrt{2} \]

    Since \(\sqrt{2}\) is irrational, the set is empty.

    \[ B = \varnothing \]

  3. C ={ x : x is an even prime number greater than 2 }

    2 is the only even prime number.

    \[ C = \varnothing \]

  4. D = {x : xΒ² = 4, x is odd}

    \[ x = \pm 2 \]

    Both values are even.

    \[ D = \varnothing \]

1.4 FINITE AND INFINITE SETS


Introduction

\[ A = \{1,\;2,\;3,\;4,\;5\} \] \[ B = \{a,\;b,\;c,\;d,\;e,\;g\} \] \[ C = \{x : x \text{ is a man living presently in the world}\} \]

Set A has 5 elements, set B has 6 elements, and set C has a large but definite number of elements.

Number of Elements

The number of elements of a set \(S\) is denoted by:

\[ n(S) \] \[ n(A) = 5 \] \[ n(B) = 6 \]
πŸ“Œ Definition 2

A set which is empty or consists of a definite number of elements is called a finite set. Otherwise, it is called an infinite set.

Examples

Days of the week:

\[ W = \{Monday,\;Tuesday,\;\ldots\} \]

W is a finite set.

Solutions of:

\[ x^2 - 16 = 0 \] \[ x^2 = 16 \] \[ x = \pm 4 \]

Finite set.

Infinite Sets

\[ \{1,\;2,\;3,\;\ldots\} \] \[ \{1,\;3,\;5,\;7,\;\ldots\} \] \[ \{\ldots,\;-2,\;-1,\;0,\;1,\;2,\;\ldots\} \]

These sets never end, hence they are infinite.

Example 6

State which of the following sets are finite or infinite :

(i) {x : x ∈ N and (x – 1) (x –2) = 0}

  1. \[ (x-1)(x-2)=0 \] \[ x=1,\;2 \]

    Finite set

    2. (ii) {x : x ∈ N and x² = 4}

  2. \[ x^2=4 \] \[ x=2 \]

    Finite set

    3. (iii) {x : x ∈ N and 2x –1 = 0}

  3. \[ 2x-1=0 \] \[ x=\frac{1}{2} \]

    Not a natural number β†’ Empty β†’ Finite

    4. (iv) {x : x ∈ N and x is prime}

  4. Set of all prime numbers β†’ Infinite

    5. (v) {x : x ∈ N and x is odd}

  5. Set of all odd natural numbers β†’ Infinite

1.5 Equal Sets

Given two sets A and B,
if every element of A is also an element of B and if every element of B is also an element of A, then the sets A and B are said to be equal.

A = B


Clearly, the two sets have exactly the same elements.

Definition 3: Two sets A and B are said to be equal if they have exactly the same elements and we write A = B. Otherwise, the sets are said to be unequal and we write A β‰  B.

We consider the following examples :
(i) Let A = {1, 2, 3, 4} and B = {3, 1, 4, 2}. Then A = B.
(ii) Let A be the set of prime numbers less than 6 and P the set of prime factors of 30. Then A and P are equal, since 2, 3 and 5 are the only prime factors of 30 and also these are less than 6.

"Repetition of elements of a set doesn't matter"

A set does not change if one or more elements of the set are repeated. For example, the sets A = {1, 2, 3} and B = {2, 2, 1, 3, 3} are equal, since each element of A is in B and vice-versa. That is why we generally do not repeat any element in describing a set.

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: xΒ² = 25},
E = {x : x is an integral positive root of the equation xΒ² – 2x –15 = 0}.

Solutions:

Example 8 Which of the following pairs of sets are equal? Justify your answer.
(i) X, the set of letters in β€œALLOY” and B, the set of letters in β€œLOYAL”.
(ii) A = {n : n ∈ Z and nΒ² ≀ 4} and B = {x : x ∈ R and xΒ² – 3x + 2 = 0}.

Solutions: (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 βˆ‰ B, A and B are not equal sets.

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