**Finite and infinite sets**

A set which consists of a finite number of elements is called a finite set otherwise, the set is called an infinite set.

**Finite Set**

[__Definition__] A set is said to be finite, if it has finite number of elements.

Example:

(i) {1, 2, 3, 5}

(ii) The letters of the English alphabet.

**Infinite Set**

[__Definition__] A set is infinite, if it is not finite.

Example:

(i) The set of all real numbers.

(ii) The points on a line.

⛲ Example 1. 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)

✍ Solution:

(i) The set of lines which are parallel to the *x*—axis is an infinite set because lines parallel to the *x*—axis are infinite in number.

(ii) The set of letters in the English alphabet is a finite set because it has 26 elements.

(iii) The set of numbers which are multiple of 5 is an infinite set because multiples of 5 are infinite in number.

(iv) The set of animals living on the earth is a finite set because the number of animals living on the earth is finite (although it is quite a big number).

(v) The set of circles passing through the origin (0, 0) is an infinite set because infinite number of circles can pass through the origin.

**lnfinite Sets and dot notation**

A set may have infinitely many elements, so We can’t list all of them. For example let *E*={all even integers greater than or equal to 1}.

We write this as *E*={2,4,6,…}, where “…” is an ellipsis, it should be read as “et cetera”.

When we place an element after the dots, as in *K*={2,4,6,…,100}, this indicates that we are talking about the finite set of even numbers greater than 1 and less than or equal to 100 (the last element on the list is 100).

[__Note__] This notation assumes we have an implicit agreement as to the formula for the remaining terms. For example *E* might have been {2, 4, 6, 12, 14, 16, 22, 24, 26, …}.

⛲ Ex2. Which of the following sets are finite and which are infinite?

(i) The set of the months of a year.

(ii) {1,2,3,…}

(iii) The set of prime numbers less than 99.

✍ Solution:

(i) It is a finite set, as there are 12 members in the set which are months of the year.

(ii) It is an infinite set, since there are infinite natural numbers.

(iii) It is a finite set, because the set is {2,3,5,7,…,97}.

**Finite and Infinite Sets**

Let *A*={1,2,3,4,5}, *B*={a,b,c,d,e,g}

and *C*={men living presently in different parts of the world}.

We observe that A contains 5 elements and *B* contains 6 elements. How many elements does *C* contain? As it is, we do not know the number of elements in C, but it is some natural number which may be quite a big number. By number of elements of a set S, we mean the number of distinct elements of the set and we denote it by n(*S*). If n(*S*) is a natural number, then *S* is non-empty finite set.

Consider the set of natural numbers. We see that the number of elements of this set is not finite since there are infinite number of natural numbers. We say that the set of natural numbers is an infinite set. The sets *A*, *B* and *C* given above are finite sets and n(*A*)=5, n(*B*)=6 and n(*C*)=some finite number.

[__Definition__] A set which is empty or consists of a definite number of elements is called __finite__ otherwise, the set is called __infinite__.

Consider some examples:

(i) Let *W* be the set of the days of the week. Then *W* is finite.

(ii) Let *S* be the set of solutions of the equation *x*^{2}-16=0. Then *S* is finite.

(iii) Let *G* be the set of points on a line. Then *G* is infinite.

When we represent a set in the roster form, we write all the elements of the set within braces {}. It is not possible to write all the elements of an infinite set within braces {} because the numbers of elements of such a set is not finite. So, we represent some infinite set in the roster form by writing a few elements which clearly indicate the structure of the set followed (or preceded) by three dots.

For example, {1,2,3,…} is the set of natural numbers, {1,3,5,7,…} is the set of odd natural numbers, {…,-3,-2,-1,0,1,2,3,…} is the set of integers. All these sets are infinite.

[

Note] All infinite sets cannot be described in the roster form. For example, the set of real numbers cannot be described in this form, because the elements of this set do not follow any particular pattern.

**Finite Set**

A set, which is empty or consists of a definite number of elements, is called a finite set. e.g.,

(i) The set {1,2,3,4} is a finite set, because it contains a definite number of elements i.e., only 4 elements.

(ii) The set of solutions of *x*^{2}=25 is a finite set, because it contains a definite number of elements i.e., 5 and -5.

(iii) An empty set, which does not contain any element is also a finite set.

The number of distinct elements in a finite set *A* is called **cardinal number** of set and it is denoted by n(*A*).

e.g., If *A*={-3,-1,8,10,13}, then n(*A*)=5.

**Infinite Set**

A set which consists of infinite number of elements is called an infinite set.

When we represent an infinite set in the roster form, it is not possible to write all the elements within braces {} because the number of elements of such a set is not finite, so we write a few elements which clearly indicate the structure of the set following by three dots.

e.g., Set of squares of natural numbers is an infinite set, because such natural numbers are infinite and it can be represented as

[

Note]

All infinite sets cannot be described in the roster form. For example, the set of real numbers cannot be described in this form, because the elements of this set do not follow any particular pattern.

⛲ Ex3. Which of the following sets are finite or infinite

(i) The set of months of a year

(ii) {1,2,3,…}

(iii) {1,2,…,99,100}

(iv) The set of positive integers greater than 100

(v) The set of prime numbers less than 99

✍ Solution:

(i) The set of months of a year is a finite set because it has 12 elements.

(ii) {1,2,3,…} is an infinite set as it has infinite number of natural numbers.

(iii) {1,2,3,…,99,100} is a finite set because the numbers from 1 to 100 are finite in number.

(iv) The set of positive integers greater than 100 is an infinite set because positive integers greater than 100 are infinite in number.

(v) The set of prime numbers less than 99 is a finite set because prime numbers less than 99 are finite in number.

**Finite and Infinite sets**

• A set is called **finite** if we can list all of its elements.

• An **infinite** set has the property that no matter how many elements we list, there are always more elements in the set that are not on our list.

• If *S* is a finite set, the symbol |*S*| stands for the number of elements of *S*.

• The set with no elements is called the empty set, and is written as Ø. Thus |Ø|=0.

• A one—element set is a set such as *S*={5} with |*S*|=1.

⛲ Ex4. 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 *x*^{2}=4}

(iii) {*x*:*x*∈ℕ and 2*x*-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=Ø. 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.

🌈 What is a Null set or Empty Set?