Finite and Infinite sets

By taking examples from real life, one is in a better position to understand concepts such as finite sets, infinite sets, and singleton sets. It is also important to point out that elements of a set can be counting numbers and if say there is a classroom and on average let us assume that there are several students in the classroom that means the set is a finite set because it can count the objects in the set. However, the set of natural numbers is rather countable or finite as there is no number and counting cannot possibly happen. On a simpler scale, consider a singleton set:

In this article, we will cover the concepts of the finite set, Infinite set, and Singleton 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 is a Set?

Sets are a foundational concept in mathematics, central to various fields such as statistics, geometry, and algebra. A set is 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…...

A set that contains only a fixed number of elements like the number of students in a particular class is known as a finite set and the number of elements in a finite set is countable. An infinite set, like a set of natural numbers, cannot be cataloged or completed since it has no bound. A single-tone set has a single component; for instance, the only chalkboard commonly found in a particular classroom. These examples will assist in illuminating these basic concepts of mathematics, or as they are also known, arithmetic skills.

Define Finite set

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

Examples: φ, {a}, {1,2,5,9}, {x: x is a person of age more than 18}

Example: A set of months in a year.
M = {January, February, March, April, May, June, July, August, September, October, November, December}

n (M) = 12

It is a finite set because the number of elements is countable.

Properties of Finite sets

The following finite set conditions are always finite.

• A subset of the Finite set
• The union of two finite sets
• The power set of a finite set

Define 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', R

{x : 2 < x < 2.1}

Properties of Infinite Sets

• The union of two infinite sets is infinite
• The power set of an infinite set is infinite
• The superset of an infinite set is also an infinite

Summary

It is important to understand that 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. The first one of the pioneered concepts is the singleton set which means a set comprising a single element and an example of this is only a chalkboard in the classroom.

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Solved Examples Based On the Finite set, Infinite set, and Singleton set:

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) 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: Which of the following is not a singleton set?

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) S={}.

Solution:

(1) set = {1}

(2) set = {2}

(3) There are two numbers that divide 12 and 20 i.e. 2 and 4. So, the set is {2, 4}.

(4) {} is a singleton set with one element .

Hence, the answer is the option (3).

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.