Finite and Infinite sets

Consider the set of all even numbers less than 100, the elements in this set is countable. Now consider the set of all natural numbers, the elements in this set is not countable as the natural numbers go on till infinity. A set with countable elements is a finite set whereas a set in which the number of elements is uncountable is infinite set.

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This Story also Contains

1. What are Finite and Infinite Sets?
2. Finite set
3. Infinite Set
4. Distinguish between Finite and Infinite Sets
5. Solved Examples Based On the Finite and Infinite sets

A finite set contains countable quantities of something, for example, how many students exist in a single class. The uncountable set can be defined as a set that has an infinity and, therefore, the number of members in the set cannot be counted like the natural numbers set that has no finite upper limit. In this article, let's look into finite and infinite sets definition and examples for finite and infinite sets

This article is about the concepts of the finite set and infinite set. This concept falls under the broader category of sets relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more.

What are Finite and Infinite Sets?

Sets are simply a collection of distinct objects, considered as a whole. These objects, called elements or members of the set, can be anything: numbers, people, letters, etc. Sets are particularly useful in defining and working with groups of objects that share common properties.

It is a well-defined collection of distinct objects and it is usually denoted by capital letters A, B, C, S, U, V…...

Consider the set of all even numbers less than 100, the elements in this set is countable. Now consider the set of all natural numbers, the elements in this set is not countable as the natural numbers go on till infinity. A set with countable elements is a finite set whereas a set in which the number of elements is uncountable is infinite set.

Finite set

A set that is empty or consists of a finite number of elements is called a finite set.

Examples: $\varphi,\{\mathrm{a}\},\{1,2,5,9\},\{\mathrm{x}: \mathrm{x}$ is a person of age more than 18$\}$

Properties of finite Sets

• A subset of the Finite set is finite
• The union of two or more finite sets is finite
• The power set of a finite set is countable

Cardinality of Finite Set

If '$a$' represents the number of elements of set $A$, then the cardinality of a finite set is $n(A)=a$. The cardinality of a finite set is a natural number or possibly $0$, it can be either.

So, the Cardinality of set $A$ of all English Alphabets is 26 because the number of elements (alphabets) is $26$.

Hence, $n(A)=26$.

Infinite Set

A set which has infinite elements is called an infinite set.

Examples: A set of all the lines passing through a point, set of all circles in a plane, set of all points in a plane, $N, Z, Q, Q^{\prime}, R$, $
\{x: 2<x<2.1\}
$

Properties of Infinite Sets

• The union of two or more infinite sets is infinite
• The power set of an uncountable infinite set is infinite
• The superset of an infinite set yeilds an infinite set

Cardinality of Infinite Sets

The cardinality of a set is $n(A)=x$, where $x$ is the number of elements of a set $A$. The cardinality of an infinite set is $n(A)=\infty$ as the number of elements is unlimited in it.

Distinguish between Finite and Infinite Sets

Note: Empty and singleton sets are finite sets.

Recommended Video Based on the Finite and Infinite Sets

Solved Examples Based On the Finite and Infinite sets

Example 1: Which of the following sets is a finite set?

1) $P=\{$ natural numbers greater than $50\}$
2) $Q=\{$ integers less than $5\}$
3) $R=\{$ whole numbers more than $10\}$
4) $\mathrm{S}=\{$ natural numbers less than $5\}$

Solution

There are only $4$ natural numbers less than $5$. So it is a finite set.

Hence, the answer is the option 4.

Example 2: Which of the following is an example of an infinite set?

1) Set of all the persons living in India.

2) Set of all the human beings living on Mars.

3) Set of all the stars in the Universe.

4) Set of the satellites of Earth.

Solution:

As there are infinite stars in the Universe, so it is an infinite set.

Note that the population of India is finite.

Hence, the answer is the option 3.

Example 3: Which of the following is not an infinite set?

1) Set of all real numbers.

2) Set of all perfect squares.

3) Set of all the divisors of $x$, where $x ∈ N$.

4) Set of all prime numbers.

Solution:

The number of divisors of a number is finite.

Hence, the answer is the option 3.

Example 4: Classify the following as finite and infinite sets?

1) Set of all natural numbers which are neither a prime nor a composite number.

2) Set of even prime numbers.

3) Set of numbers that divide $12$ and $20$.

4) $\mathrm{S}=\{\phi\}$.

Solution:
(1) set $=\{1\}$ - finite set
(2) set $=\{2, 3, 5, 7, 13,...... \}$ - infinite set
(3) Set - $\{2,4\}$ - finite set
(4) $\left\{\phi{\}}\right.$ - finite set

Example 5: Which of the following sets is a finite set?
1) Set of all points in a plane.
2) Set of all points on a line segment.
3) Set of all lines in a plane.
4) Set of all circles passing through three non-collinear points.

Solution

In this question,

Number of geometrical points and lines in a plane are infinite. Also, the number of points in a line segment is infinite.

However, there is only one circle passing through three non-collinear points, so it is a finite set (as it has one element).

Hence, the answer is the option 4.

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