Introduction

This book is intended to organize my thoughts on mathematics and to provide a reference for the concepts that I have learned.

What is Mathematics?

It is very difficult give a precise definition of mathematics. It is a very broad field that encompasses many different areas of study. For this reason, I will give the definition that I like the most. First i want to define what mathematics is not.

Mathematics is not only about numbers: A lot of people think that mathematics is only about numbers. This is not true. For instance, logic, abstract algebra, topology, and geometry are areas of mathematics and they can be studied without using numbers.

Although, mathematics is not only about the things that are listed above, it is also about them. So, what is mathematics? Mathematics is an approach to understand the world around us by modeling them. And these models are actually abstractions of the real world. This is why mathematics is so powerful. It is not about the real world, it is about the models of the real world.

For this reason, I believe that the most important property of mathematics is abstraction. Here are some quotes that support this idea:

Mathematics is the art of giving the same name to different things.

Henri Poincaré

Mathematics is the art of reasoning about quantitative relations between abstract structures.

Paul Erdős

Mathematics is the science of patterns.

Keith Devlin

Mathematics is the science of patterns, and we study patterns in the abstract.

Ron Aharoni

This definition, I think captures the essence of mathematics. Another approach that I follow generally to define a thing

What is abstraction?

Abstraction is a process of removing details from a concept in order to focus on the essential properties of that concept. Lets look some examples of abstraction:

A word apple is an abstraction of the concept of an apple. We are removing the details of the apples and focusing on the essential properties of an apple. Lets imagine that we have two apples in front of us. One of them is red and the other one is green. We can say that the red apple and the green apple are different. But, if we abstract the concept of an apple, we can say that they are the same and both of them are apples. This is the power of abstraction.

Definitions

A definition is where everything starts. It is the most basic concept in mathematics. A definition is a statement that gives the meaning of a term. It is a way to give a name to a concept. Without giving names to concepts, it would be very difficult to talk about them. For this reason, definitions are very important in mathematics. If you think, you could not grasp a topic you should first check the definitions of the terms that are used in that topic. Without understanding the definitions, it is impossible to understand the topic. You should really understand the definitions.

Lets look some examples of definitions:

A natural number is either zero or a successor of a natural number.

A number is said to be even if it is divisible by 2.

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers.

Propositions

A proposition is a statement that is either true or false.

Axioms

An axiom is a statement that is assumed to be true without a proof. Axioms are the foundation of mathematics. They are the building blocks of a mathematical structure. Axioms are the starting point of a mathematical theory.

Proofs

Theorems

Lemmas and Corollaries

Conjectures

Fundamental Concepts

Sets

Functions

Relations

Proofs

Fundamental Concepts

Sets

Functions

Relations

Proofs

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:

$X = {x ∣ x is an even number}$

or in more general form:

$X = {x ∣ P (x)}$

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 $A \neq = B$ .

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:

$A \cup B = {x ∣ x \in A or x \in B}$

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:

$A \cap B = {x ∣ x \in A and x \in B}$

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:

$A - B = {x ∣ x \in A and x \in / B}$

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:

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

Relations and Functions

Cardinality

Countable and Uncountable Sets

Infinite Sets

Axiomatic Set Theory

Russell's Paradox

Relations

Definition

Symmetry

Reflexivity

Transitivity

Equivalence Relations

Partial Orders

Hasse Diagrams

Lattices

Functions

Definition

Injective Functions

Surjective Functions

Bijective Functions

Composition of Functions

Inverse Functions

Cardinality of Sets and Functions

TODO: Recursive Functions

First Order Logic

Logic is the study of reasoning, a fundamental part of human cognition. Logic is used almost everywhere in mathematics. Being sure about reasoning is an indispensable part of logic.

One of the most important questions of philosophy is "What is the source of truth?". This question may answered differently by different people.

Source of Truth

TODO: There are different logical systems.

Formal Systems

A formal system is consists of;

A formal language which is describes the well-formed formulas of the system. It is also called the syntax of the system.
A set of rules that are used to manipulate the formulas of the language. These rules are used to derive new formulas from the given formulas. These rules are also called the semantics of the system.

A formal language is constructed by specifying an alphabet and a set of rules that are used to build formulas. Here are the definitions:

An alphabet $A$ is a set of symbols.
A string or formula of an alphabet $A$ is an ordered sequence of symbols from the alphabet and is denoted by $A^{*}$ .
A well formed formula (wff) is a formula that is formed according to the (syntax) rules of the language.

TODO: Truth semantics and proof semantics.

Propositional Logic

Propositional logic is a branch of logic that deals with propositions. A proposition is a statement that is either true or false. Propositional logic is also called sentential logic or statement logic. Propositional logic is the simplest form of logic and is used to study the internal structure of propositions. In this section, propositional logic is studied in a formal way. First a formal system is defined and then the syntax and different semantics of propositional logic are studied.

Syntax

Propositions are statements that are either true or false. Formal propositional logic is a branch of logic that deals with only propositions in a formal way. To study a subject formally, it is necessary to represent the subject in a symbolic form. For this reason, an alphabet and a formula are defined. An alphabet is a set of symbols that are used to build formulas. A formula (sentence) is a string of symbols from the alphabet that represents a proposition.

Propositions may be constituted by using natural language, but natural language is not precise enough to represent propositions formally. For this reason, in propositional logic, the internal structure of propositions is not considered instead each proposition is considered as a whole entity. Propositions are represented by symbols such as $P_{1}$ , $P_{2}$ , $P_{3}$ , etc. These symbols are called propositional variables, atomic propositions, or prime formulas.

To study more complicated propositions, it is necessary to be able to build more complex propositions from simpler ones. This is done by using logical connectives. Logical connectives are symbols that are used to combine propositions to form more complex propositions. The most common logical connectives are:

Negation ( $\neg P$ )
Conjunction ( $P \land Q$ )
Disjunction ( $P \lor Q$ )
Implication ( $P ⟹ Q$ )
Biconditional ( $P ⟺ Q$ )

The sentences formed by using logical connectives are called compound propositions or compound formulas.

The alphabet of propositional logic is defined as follows;

$A = {\neg, (,), \land, \lor, ⟹, ⟺, P_{0}, P_{1}, \dots}$

A well-formed formula (wff) of propositional logic;

An atomic formula is a well-formed formula. (e.g. $P_{0}$ , $P_{1}$ , $\dots$ )
If $ϕ$ is a well-formed formula, then $\neg ϕ$ is a well-formed formula.
If $ϕ$ and $ψ$ are well-formed formulas, then $(ϕ \land ψ)$ , $(ϕ \lor ψ)$ , $(ϕ ⟹ ψ)$ , and $(ϕ ⟺ ψ)$ are well-formed formulas.

TODO: Omiting the parentheses.

Truth Semantics

Well-formed formulas does not have any meaning by themselves. They are just valid and meaningful strings of symbols. For example $P_{0} \land P_{1}$ is a well-formed formula but it does not have any meaning by itself. To give meaning to it, it is necessary to define;

The derivation rules for the truth values of compound propositions.
The truth values of atomic propositions.

Truth semantics of propositional logic is defined by using truth tables. A truth table is a table that shows the truth values of a compound proposition for all possible truth values.

$ϕ$	$ψ$	$\neg ϕ$	$ϕ \land ψ$	$ϕ \lor ψ$	$ϕ ⟹ ψ$	$ϕ ⟺ ψ$
F	F	T	F	F	T	T
F	T	T	F	T	T	F
T	F	F	F	T	F	F
T	T	F	T	T	T	T

Proof Semantics

Soundness and Completeness

References

Kleene, S. C. (1967). Mathematical logic. Dover Publications.

Predicate Logic

Algebra

Algebra is the study of mathematical symbols and the rules for manipulating these symbols. Historically, algebra was emerged from the need to solve equations and understand the properties of numbers. Algebra is a broad field that encompasses various subfields such as elementary algebra, linear algebra, abstract algebra, and universal algebra.

The word algebra is derived from the Arabic word "al-jabr", which means "reunion of broken parts." The term was first used by the Persian mathematician Al-Khwarizmi in his book "Kitab al-Jabr wa al-Muqabala". After that, Omar Khayyam, defined algebra as the science of solving equations.

One of the most influential results of algebra is the development of abstract algebra, which studies algebraic structures in a more general setting. Abstract algebra provides a unified framework for studying various algebraic structures such as groups, rings, fields, and vector spaces. These algebraic structures are essential in many areas of mathematics and science, including number theory, geometry, and physics.

An algebraic structure is a set equipped with one or more operations that satisfy certain properties. The study of algebraic structures and their properties is the central theme of algebra. Some examples of algebraic structures include groups, rings, fields, and vector spaces.

Linear Algebra

Universal Algebra

Groups

One of the most fundamental algebraic structures is a group. Generally, groups are used to study the symmetries of objects and the solutions to equations.

A group G is a set with a binary operation * that satisfies the following properties:

Closure: For all a, b in G, a * b is also in G.

Associativity: For all a, b, c in G, (a * b) * c = a * (b * c).

Identity Element: There exists an element e in G such that for all a in G, a * e = e * a = a.

Inverse Element: For all a in G, there exists an element a' in G such that a * a' = a' * a = e.

Group Examples

(Z, +) is a group where Z is the set of integers and + is the addition operation.

The identity element is 0.
Every element a has an inverse -a.
The operation is associative.
The set is closed under addition.

(Q, +) is a group where Q is the set of rational numbers and + is the addition operation.

Theorems

Theorem: The identity element of a group is unique. Proof: Suppose there are two identity elements e and e'. Then, e = e * e' = e'.

Theorem: The inverse element of a group is unique. Proof: Suppose there are two inverse elements a' and a'' for an element a. Then, a' = a' * e = a' * (a * a'') = (a' * a) * a'' = e * a'' = a''.

Theorem (Cancelation Law): If a * b = a * c, then b = c. Proof: Multiply both sides by a' on the right. a * b = a * c a * b * a' = a * c * a' b * a' = c * a' b = c

Subgroups

Definition (Subgroup): A group (H, *) is a subgroup of a group (G, *) if H is a subset of G and H is a group under the same operation *.

TODO: Trivial subgroup TODO: Proper subgroup TODO: Cyclic subgroup TODO: Order of a subgroup TODO: Lagrange's theorem,

Math Notes