When we explore the vast world of mathematics, the **probability And Statistics, **we find a fundamental concept: subsets. These mathematical entities play a crucial role in various branches of knowledge, providing a solid foundation for understanding and analyzing larger and more complex assemblies.

In simple terms, a subset refers to a mathematical set that contains selected elements. **from another larger set**. By extracting specific elements, subsets allow us to examine and study particular features, giving us a deeper understanding of the structure and properties of the original set.

Subsets are used in set theory, probability, and statistics, help define inclusion relationships, and **conduct joint operations**. Subsets play a vital role in data analysis and predictive modeling.

## Meaning of subset in mathematics

In simple terms, a subset refers to a set that **contains elements selected from another larger set**. By extracting specific elements, subsets allow us to examine and study particular features, giving us a deeper understanding of the structure and properties of the original set.

The mathematical notation used to represent a subset is **the symbol ⊆.** If A is a subset of B, it is written as **A ⊆ B**. Also, the symbol ⊂ is used to indicate that A is a proper subset of B, which means that A is a subset of B, but is not equal to B.

For example, consider two sets: **A = {1, 2, 3} and B = {1, 2, 3, 4, 5}.** In this case, A is a subset of B because all the elements of A (1, 2, and 3) are also present in B. Therefore, we can write A ⊆ B.

It is necessary to mention that a set **it is also considered a subset of itself. **For example, in the above case, A is a subset of itself because all the elements of A are in A.

### What is subset in probability and statistics?

**In probability,** a subset is used to define an event. An event is a set of possible outcomes in a probabilistic experiment.

For example, consider rolling a six-sided die. The set of possible outcomes is** {1, 2, 3, 4, 5, 6}.** If we consider the event ‘get an even number’, the corresponding subset would be {2, 4, 6}.

In terms of notation, it can be denoted as** A = {2, 4, 6}**where A is a subset of the set of possible outcomes.

**in statistics**subsets are used to represent specific data sets within a sample or population.

For example, if you are studying the height of people in a city, you can select subsets based on specific criteria, such as ‘people under 30’ or **‘persons over 50 years of age’.**

### What is subset in algebra?

**in algebra**, a subset refers to a set that contains elements that are also present in another, larger set. However, in algebra, subsets can also have an additional relationship with respect to algebraic operations and properties.

Given a set A, a subset B of A is defined as a set where all elements of B **are also elements of A**. In terms of notation, it is written as B ⊆ A. This means that any element that belongs to B also belongs to A.

## What is a proper subset?

In mathematics, a proper subset is a subset that contains some, **but not all**, the elements of another larger set. In other words, if all the elements of a set A are also present in a set B, but there are also elements in B that are not in A, then A is said to be a proper subset of B.

## What is an improper subset?

An improper subset is a subset that contains **all the elements of another set**, including the original set itself. That is, if all the elements of a set A are also present in a set B, and also B contains all the elements of A, then A is said to be an improper subset of B.

For example, consider two sets: **A = {1, 2, 3} and B = {1, 2, 3, 4, 5}**. In this case, A is an improper subset of B because all the elements of A (1, 2, and 3) are present in B, and B contains all the elements of A. Therefore, we can write A ⊆ B.

## How is a subset represented?

A subset is represented **using the symbol ⊆ or ⊂. **These symbols indicate that a set is a subset of another set.

The symbol ⊆ is used to **represent a subset**, including the possibility that the original set and the subset are equal. If A is a subset of B or if A is equal to B, it is written as A ⊆ B.

On the other hand, the symbol ⊂ is used to **represent a proper subset,** which means that the original set and the subset are not equal. If A is a proper subset of B, it is written as A ⊂ B.

## Examples of subsets in mathematics

Some **examples of subsets **in mathematics are:

consider the whole **A = {1, 2, 3}** and the set B = {1, 2, 3, 4, 5}. In this case, A is a subset of B because all the elements of A are also in B. Therefore, A ⊆ B.

If we have the set **C = {a,b,c}** and the set D = {a, b, c, d}, we can say that C is a subset of D, since all elements of C are also in D. As a result, C ⊆ D.

If we consider the set **E = {x | x **is a positive integer less than 5}, then E is a subset of the set of natural numbers. We can write it as E ⊆ N.