# How Mathematicians Quantify the Universe using Set Theory

### Discussing Set Theory - Notations, Cardinality, Operations, Subsets, Supersets, Power sets, & Applications

Last week, we discussed how mathematicians define *‘Axioms’,* the fundamental mathematical truths using mathematical notation and deduction. In this week’s article, we continue this journey by discussing another fundamental concept in math - *Sets.*

A **set** is a collection of distinct objects, known as **elements** or **members**, grouped together based on a **common property**. These objects can be anything: numbers, letters, colours, or even other sets.

Upon initial observation, sets may appear unassuming, yet they indubitably assume a pivotal role in delineating numerous foundations of mathematics. Moreover, their profound significance extends to the comprehension of renowned paradoxes such as the Russel's Paradox.

## Defining a Set

Sets are typically denoted by **capital letters**, and their elements are enclosed within **curly braces**.

### Example

Set A = {1, 2, 3, 4, 5}

Set B = {apple, orange, banana, peach}

**In set theory, an element either belongs to a set (is an element of) or does not belong to it.** We use the symbol **∈ **(pronounced as, *‘belongs to’*) to represent membership and **∉ **(pronounced as, *‘does not belong to’*) for non-membership.

### Example

2 ∈ A (2 is an element of set A)

6 ∉ A (6 is not an element of set A)

Understanding sets and their operations provides a strong foundation for tackling more complex mathematical problems and contributes to various fields such as computer science, statistics, and theoretical physics.

## Set Notations

Set theory employs different notations to represent sets in concise and systematic ways. Additionally, understanding the **cardinality** of sets, or the size of their elements, plays a crucial role in various mathematical applications.

### Roster Notation

Roster notation is a straightforward method of listing the elements of a set **explicitly**, separated by commas, and enclosed within **curly braces**. This notation is suitable for finite sets with a small number of elements.

#### Example

Consider a set *X* that contains the first four prime numbers:

X = {2, 3, 5, 7}

### Set-Builder Notation

Set-builder notation defines a set by specifying the **properties or characteristics** that its elements must satisfy. The general form is *{x | P(x)},* where *"x"* represents the elements, and *"P(x)"* is the condition or property that the elements must fulfil.

#### Example

Let's define a set *Y* of even natural numbers less than ten using set-builder notation:

Y = {x | x is a natural number, x is even, and 0 < x < 10}

Y = {2, 4, 6, 8}

## Cardinality of Sets

The cardinality of a set, denoted by *|A|,* represents the **number of elements** in the set A.

### Finite Cardinality

For finite sets, determining the cardinality is straightforward. **The cardinality of a set A is the total number of elements in A.**

#### Example

Let's revisit the set *X* from the roster notation example:

X = {2, 3, 5, 7}

|X| = 4 (since there are four elements in set X)

### Infinite Cardinality

Infinite sets have an uncountable number of elements. They are classified as either **countably infinite** or **uncountably infinite**.

#### Countably Infinite

A set is countably infinite if its elements can be put into a **one-to-one correspondence with the natural numbers** (1, 2, 3, ...). Such sets have the same cardinality as the set of natural numbers.

**Example**

The set of all natural numbers is countably infinite:

N = {1, 2, 3, ...}

|N| = ∞ (infinity)

#### Uncountably Infinite

A set is uncountably infinite if it has a **higher cardinality than the set of natural numbers**. An example of an uncountably infinite set is the **set of real numbers**.

## Basic Set Operations

Sets serve as the building blocks of set theory, and understanding their operations is essential for solving mathematical problems and exploring various applications.

### Union ( ∪ )

The union of two sets *A* and *B*, denoted as *A ∪ B,* creates a new set that contains all the elements present in either *A* or *B,* or both. In other words, i**t combines the elements from both sets while removing duplicates**.

#### Example

Let's consider two sets:

A = {1, 2, 3}

B = {3, 4, 5}

The union of sets A and B is:

A ∪ B = {1, 2, 3, 4, 5}

### Intersection ( ∩ )

The intersection of two sets *A* and *B,* denoted as *A ∩ B,* creates a new set containing **only the elements that are present in both sets ****A**** and ****B**.

#### Example

Using the same sets as before:

A = {1, 2, 3}

B = {3, 4, 5}

The intersection of sets A and B is:

A ∩ B = {3}

### Difference ( \ )

The difference between sets *A* and *B,* denoted as *A \ B,* creates a new set containing **elements that are in set ****A**** but not in set ****B**.

#### Example

Considering the same sets again:

A = {1, 2, 3, 4}

B = {3, 4, 5}

The difference of sets A and B is:

A \ B = {1, 2}

### Complement ( ' or ∁ )

The complement of a set *A*, denoted as *A'* or *∁(A)*, consists of **all the elements that do not belong to ****A**** within a specific universal set ****U**.

#### Example

Let's consider a universal set U as the set of all natural numbers, and set A as:

A = {2, 4, 6}

The complement of set A is:

A' = {1, 3, 5, 7, ...} (All natural numbers except 2, 4, and 6)

### Disjoint Sets

Two sets are disjoint if their intersection is an empty set, meaning **they have no elements in common.**

#### Example

Consider two sets:

A = {1, 2, 3}

B = {4, 5, 6}

Since A ∩ B = {} (an empty set), sets A and B are disjoint.

Mastering these basic set operations allows mathematicians, scientists, and computer programmers to model and solve a wide range of problems.

## Subsets and Supersets

In set theory, subsets and supersets play a crucial role in understanding the relationships between different sets. These concepts help us analyse how sets are related to one another and are fundamental to various mathematical proofs and real-world applications.

### Subsets

A set *A* is considered a subset of another set *B* if all the elements of *A* are also elements of *B*. In other words, **every element in A is contained within B.** We denote this relationship as A ⊆ B.

#### Example

Let's consider two sets:

A = {1, 2, 3}

B = {1, 2, 3, 4, 5}

Since all the elements of A (1, 2, and 3) are also elements of B, we can say that A is a subset of B: A ⊆ B.

### Proper Subset

A subset *A* is called a proper subset of B if *A* is a subset of *B* but not equal to *B.* In other words, **all elements of A are also elements of B, but B has additional elements not in A.** We denote this relationship as A ⊂ B.

#### Example

Using the same sets as before:

A = {1, 2, 3}

B = {1, 2, 3, 4, 5}

Since A is a subset of B and A is not equal to B, we can say that A is a proper subset of B: A ⊂ B.

### Supersets

Conversely, a set *B* is considered a superset of another set *A* if all the elements of *A* are also elements of *B.* In other words, **B**** contains every element of *** A.* We denote this relationship as B ⊇ A.

#### Example

Using the same sets as before:

A = {1, 2, 3}

B = {1, 2, 3, 4, 5}

Since all the elements of A (1, 2, and 3) are also elements of B, we can say that B is a superset of A: B ⊇ A.

### Proper Superset

A superset *B* is called a proper superset of *A* if *B* is a superset of *A* but not equal to *A.* In other words, **B**** contains all elements of ****A**** and has additional elements not in *** A.* We denote this relationship as B ⊃ A.

#### Example

Using the same sets as before:

A = {1, 2, 3}

B = {1, 2, 3, 4, 5}

Since B is a superset of A and B is not equal to A, we can say that B is a proper superset of A: B ⊃ A.

## Power Sets

In set theory, **the power set of a set** is a fundamental concept that enables us to explore **the collection of all possible subsets of a given set.** The power set reveals interesting properties and is essential for understanding the combinatorial aspects of sets.

### Definition of Power Set

Given a set *A,* the power set of *A,* denoted as *P(A),* is the **set that contains all possible subsets of ****A,**** including the empty set and ****A**** itself.**

#### Example

Let's consider a set A = {1, 2}.

The power set of A, P(A), consists of the following subsets:

P(A) = { {}, {1}, {2}, {1, 2} }

**Note** that the empty set {} and the set A = {1, 2} are both included in the power set.

### Cardinality of Power Set

The cardinality of the power set *P(A)* is related to the number of elements in the original set *A*. **If ****A**** contains ****n**** elements, then the power set ****P(A)**** contains ****2^n**** elements.**

#### Example

If A = {1, 2}, as shown earlier, the power set P(A) has 2^2 = 4 elements.

### Intuition Behind Power Set

The power set is a way to explore **all the possible combinations of elements that can be chosen from a given set.** It includes every possible arrangement, including the sets with no elements (the empty set) and the set with all elements (the original set itself).

#### Example

For a larger set A = {1, 2, 3}, the power set P(A) would contain 2^3 = 8 elements:

P(A) = { {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} }

The power set helps us understand the various ways we can form subsets and assists in counting the total number of possible combinations.

### Applications of Power Set

The power set finds applications in various areas, including combinatorics, probability, computer science, and discrete mathematics. For example, in computer science, the power set is used to generate all possible subsets of a given set, which is useful for tasks like solving the subset sum problem or generating combinations in algorithms.

## The Beauty of Set Theory

Set theory stands as a cornerstone of modern mathematics, laying the foundation for various branches and shaping our understanding of mathematical structures and relationships. From the study of finite collections to the exploration of infinite sets, set theory has revealed deep and elegant truths about the nature of mathematical objects.

### Universal Language of Mathematics

Set theory serves as a universal language that unifies diverse mathematical concepts and provides a systematic framework for understanding structures, relationships, and operations. By defining sets and their interactions, set theory enables mathematicians to develop a coherent language to express complex ideas.

### Logical Reasoning and Proofs

Set theory's axiomatic approach and precise definitions have revolutionised mathematical reasoning and proof techniques. The rigorous nature of set theory allows mathematicians to construct solid arguments and investigate the properties and consequences of various axioms.

### Exploring Infinity

The study of infinite sets introduces us to a realm of unparalleled beauty and richness. Concepts like the uncountable infinity of the real numbers and the Continuum Hypothesis challenge our intuition and open doors to philosophical inquiries about the nature of infinity itself.

### Applications and Impact

Beyond its theoretical elegance, set theory finds diverse applications in different fields, such as computer science, physics, engineering, and social sciences. From database management to cryptography, set theory's principles underpin critical aspects of modern technology and scientific advancements.

### The Power of Abstraction

Set theory exemplifies the power of abstraction in mathematics. By distilling complex structures into simple sets and operations, mathematicians gain a clearer and more profound understanding of underlying patterns and principles.

In the coming weeks, we will be using the base that we have laid in this article to build further upon the more complex problems in Set theory and their applications. If you haven’t already, don’t forget to subscribe to receive a new article every week sent straight to your inbox.