Naive Set Theory

Sets are the one of the most fundamental concepts in mathematics. Before diving deep into the set theory, we will start with a very non-rigorous introduction to the set theory.

Classically, a set is collection of things. These things are called elements or members of the set. For example, the set of all natural numbers is a set. A single natural number is an element of this set. Generally we are representing sets with capital letters and elements with small letters. For example, we can say that $$x\in X$$

Definition of a Set

We can define a set in two different ways. The first one is just listing the elements of the set. For example, the set of all possible outcomes of a dice roll can be defined as:

$$X=\{1,2,3,4,5,6\}$$

The second way to define a set is by using a property that the elements of the set satisfy. For example, the set of all even numbers can be defined as:

$$X=\{x\mid x\text{ is an even number}\}$$

or in more general form:

$$X=\{x\mid P(x)\}$$

where $P(x)$ is a property that the elements of the set satisfy.

Set Equality

Two sets are equal if they have the same elements. For example, the sets $A=\{1,2,3\}$ and $B=\{3,2,1\}$ are equal because they have the same elements. We can write this as $A=B$. If two sets are not equal, we can write this as $A\ne B$.

Subsets

A set $A$ is a subset of a set $B$ if every element of $A$ is also an element of $B$. We denote this as $A\subseteq B$. Formally, we can define the subset relation as follows:

$$A\subseteq B⟺\forall x\in A,x\in B$$

Set Operations

A union of two sets $A$ and $B$ is a set that contains all the elements that are in $A$ or in $B$ or in both. We denote the union of two sets $A$ and $B$ as $A\cup B$. Formally, we can define the union of two sets as follows:

$$A\cup B=\{x\mid x\in A\text{ or }x\in B\}$$

An intersection of two sets $A$ and $B$ is a set that contains all the elements that are in $A$ and in $B$ at the same time. We denote the intersection of two sets $A$ and $B$ as $A\cap B$. Formally, we can define the intersection of two sets as follows:

$$A\cap B=\{x\mid x\in A\text{ and }x\in B\}$$

A difference of two sets $A$ and $B$ is a set that contains all the elements that are in $A$ but not in $B$. We denote the difference of two sets $A$ and $B$ as $A-B$. Formally, we can define the difference of two sets as follows:

$$A-B=\{x\mid x\in A\text{ and }x\notin B\}$$

A cartesian product of two sets $A$ and $B$ is a set that contains all the possible pairs of elements where the first element is from $A$ and the second element is from $B$. We denote the cartesian product of two sets $A$ and $B$ as $A\times B$. Formally, we can define the cartesian product of two sets as follows:

$$A\times B=\{(a,b)\mid a\in A\text{ and }b\in B\}$$

Relations and Functions

Cardinality

Countable and Uncountable Sets

Infinite Sets

Axiomatic Set Theory

Russell's Paradox