# Linear Subspace

DEFINITION1: Let $K$ be a field, $X$ be a $K$-vector space and $M\subset X$. If $M$ is also a $K$-vector space, then $M$ is called a linear subspace or vector subspace (or shortly subspace) of $X$.

PROPOSITION1: Let $K$ be a field, $X$ be a $K$-vector space and $M\subset X$. $M$ is a subspace of $X$ if and only if

a) $\theta\in M$,

b) $x+y\in{M}$ for all $x,y\in{M}$,

c) $\lambda{x}\in{M}$ for all ${\lambda}\in{K}$ and ${x}\in{M}$.

# Vector Space

DEFINITION1: Let $K$ be a set with at least two distinct elements, $+:K\times{K}\rightarrow{K}$ and $\cdot:K\times{K}\rightarrow{K}$ be two functions. If the following conditions hold, then $\left( K,+,\cdot \right)$ is called a field:

F1) $\left( a+b \right) +c=a+ \left( b+c \right)$ for all $a,b,c\in{K}$,

F2) $a+b=b+a$ for all $a,b\in{K}$,

F3) there exists ${0}\in{K}$ such that $a+0=a$ for all $a\in{K}$,

F4) for each $a\in{K}$, there exists $b\in{K}$ such that $a+b=0$,

# Construction of The Real Numbers

DEFINITION1: The set $\mathbb{R}$ with at least two distinct elements and satisfying the following five axioms is called the set of real numbers and each element of $\mathbb{R}$ is called a real number:

The function $+:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ defined as $\left( x,y \right)\to x+y\in \mathbb{R}$ for each $\left( x,y \right)$ in $\mathbb{R}\times \mathbb{R}$ satisfies the following properties:

I${{}_{1}}.\,\forall a,b\in \mathbb{R},a+b=b+a$,

I${{}_{2}}.\,\forall a,b,c\in \mathbb{R},a+(b+c)=(a+b)+c$,

# Function

DEFINITION1: Let $X$ and $Y$ be two sets and $f\subset{X\times{Y}}$ be a relation. If the following two conditions are provided, then the relation $f$ is called a “function” with domain $X$ and codomain $Y$ and denoted by $f:X\to{Y}$ or $X\stackrel{f}{\rightarrow}{Y}$.

1. $\forall{x}\in{X}, \exists{y}\in{Y}: (x,y)\in{f}$,

2. $(x,y),(x,y')\in{f}\Rightarrow{y=y'}$.

Henceforth, when we write $f:X\to{Y}$, we will consider that “$f$ is a function from $X$ to $Y$”.

As is seen from the definition, a function is a relation mapping each element in the domain to a unique element in the codomain. So, the notation $y=f(x)$ is generally used instead of the notations $xfy$ and $(x,y)\in{f}$ and read “$x$ maps to $y$” or “$x$ maps to $f(x)$”. The notation $f(x)$ is read “$f$ of $x$”. Each element of the domain is called an “argument” and for each $x$ in the domain, the corresponding unique element $y$ in the codomain is called “the function value at $x$”, “output $f$ for an element $x$” or “the image $x$” under the function $f$. The set defined as $\{f(x)\:|\:x\in{X}\}\subset{Y}$ is called the “image” or the “range” of $f$. Sometimes a function is called a “map” or a “mapping”.

# Equivalence Relation

DEFINITION1: Let $X$ be a set and $R\subset{X\times{X}}$. If the relation $R$ is reflexive, symmetric and transitive, then the relation $R$ is called an "equivalence relation" and denoted by $R=\sim$ in general.

DEFINITION2: Let $\sim$ be an equivalence relation over a set $X$ and $a\in{X}$. The set defined as $\{x\in{X}\:|\:a\sim{x}\}\subset{X}$ is called the “equivalence class” of $a$ under $\sim$ and denoted by $\overline{a}$, $[a]$ or $[a]_{\sim}$. Since $a\sim{a}$ for every $a\in{X}$, then $a\in{\overline{a}}$. So the equivalence class $\overline{a}$ is non-empty for every $a\in{X}$. The family of all the equivalence classes of the relation $\sim$ is called the “quotient set” of $X$ by $\sim$ and denoted by ${^X}/{_\sim}$

I.e.,

${^X}/{_\sim}=\{\overline{a}\:|\:a\in{X}\}\subset{\mathbf{P}(X)}$.

DEFINITION3: Let $\sim$ be an equivalence relation over a set $X$ and $a,b\in{X}$. If $b\in{\overline{a}}$, then $b$ is called a “representative class” of the equivalence class $\overline{a}$.

# Partial Order Relation

DEFINITION1: Let $X$ be a set and $R\subset{X\times{X}}$. If the relation $R$ is reflexive, antisymmetric and transitive, then the relation $R$ is called a "partial order relation" and denoted by $R=\le$ in general. If "$\le$" is a partial order relation over a set $X$, then $(X,\le)$ is called "partially ordered set" or shortly "poset".

DEFINITION2: Let $x$ and $y$ are elements of a partially ordered set $X$. If it holds “$x\le{y}\lor{y\le{x}}$”, then $x$ and $y$ are called “comparable”. Otherwise they are called “incomparable”.

DEFINITION3: If $x$ and $y$ are comparable for all $x,y$ in a partially ordered set $(X,\le)$, then the relation $\le$ is called a “total order” and the set $X$ is called a “totally ordered set” or “linearly ordered set”.

DEFINITION4: Let $(X,\le)$ be a partially ordered set and $A\subset{X}$. If $(A,\le)$ is a totally ordered set, then $A$ is called a “chain” in $X$.

DEFINITION5: Let $(X,\le)$ be a partially ordered set and $A\subset{X}$. If there exists an element $a^{*}\in{A}$ satisfying $a\le{a^{*}}$ for all $a\in{A}$, then $a^{*}$ is called the maximum of $A$, and if there exists an element $a_{*}\in{A}$ satisfying $a_{*}\le{a}$ for all $a\in{A}$, then $a_{*}$ is called the minimum of $A$. The minimum and the maximum of $A$ are denoted by $\min{A}$ and $\max{A}$ respectively.

# Relation

DEFINITION1: Let $X$ and $Y$ be two sets. Any subset of the cartesian product $X\times{Y}$ is called a relation with domain $X$ and codomain $Y$. While some sources are giving the definition of relation, they assume $X,Y\ne{\varnothing}$ and it’s said the emptyset being a subset of $X\times{Y}$ isn’t a relation. However, the assumption “the emptyset is a relation” is not a problem for any branch of the mathematics. On the contrary, the assumption “the emptyset is a relation” plays an important role in some branch of the mathematics.

If $X$ and $Y$ are two sets with $n$ and $m$ elements respectively, then the cartesian product $X\times{Y}$ has $n.m$ elements. Since a relation with domain $X$ and codomain $Y$ is an element of the power set $\mathbf{P}(X\times Y)$ and the number of the elements of the power set of a set with $k$ elements is $2^{k}$, then the number of all the relations with domain $X$ and codomain $Y$ is $2^{n.m}$. If $X,Y\ne{\varnothing}$ and at least one of $X$ and $Y$ is infinite set, then the number of all the relations with domain $X$ and codomain $Y$ is also infinity.

Let $R\subset{X\times{Y}}$ be a relation not being the emptyset. The statement $(x,y)\in{R}$ is read “x is R-related to y” and is denoted by $xRy$ or $R(x)=y$.

EXAMPLE1: Let $X=\{a,b,c\}$ and $Y=\{1,2\}$. Since $X$ has 3 elements and $Y$ has 2 elements, the number of all the relations with domain $X$ and codomain $Y$ is $2^{2.3}=2^{6}=64$. We can give some of these $64$ relations:

$R_1=\varnothing$,

# Index Set

Let $X$ be a set. A set is called an “index set” of the set $X$ if $X$ and that set are equipollent. I.e., a set $I$ is called an index set of $X$ if there exists a bijective function (injective and surjective) between $I$ and $X$. As is clear from the definition, the number of all the index sets of a set may be more than one, even infinite. Besides, we can say from the definition: there exists at least one index set of any set. Because, the function $I_{X}:X\to{X}$ is bijective. This is a trivial example since the set $X$ is indexed by itself. If we indexed every set with itself, the indexing operation would be meaningless. We must give a reinforcement example: Let be $X=\{\diamondsuit, \heartsuit, \clubsuit, \spadesuit\}$. We will give three index sets for this set: The sets $I_1=X$, $I_2=\{1,2,3,4\}$ and $I_3=\{a,b,c,d\}$ can be given as index sets of $X$. (Can be given more index sets for this set). The type of index set that is widely used by the mathematicians is $I_{2}$ since it one by one counts the elements of $X$. In general, $I=\{1,2,\dots,n\}$ can be chosen as an index set for a set with $n$ elements. An index set of a set is directly associated with the cardinality of that set. Since the cardinality relation on any family of sets is an equivalence relation, an index set of a set can be actually considered as the most reasonable “representation of class” of the equivalence class of a set. For example, the most reasonable index set of any countable set is naturally the set of the natural numbers. $\mathbb{R}$, $[0,1]$ or $(0,1)$ is widely used as an index set for a set that is equipollent with the set of the real numbers.

By the end of the 19th century, the mathematicians had called “set” a collection of any objects. For example, the set of the natural numbers, the set of integers, the set of even numbers, the set of the real numbers, the set of any sets, the set of all the sets. We can give further similar examples. All the mathematicians had no doubt about that the unique condition to be a set was to gather any objects by the time Bertrand Russell’s paradox emerged. Russell had proved that when the term "set" is defined as “a collection of any objects”, there emerges a paradox in the set theory . Now, let’s examine Russell’s paradox and its proof: Assume that a collection of any objects is a set. In that case, the collection of all the sets is a set. We denote this set by $X$. Hence, any set is an element of the set $X$ i.e., if $A$ is a set, then $A\in{X}$. Since $X$ is also a set, then $X\in{X}$. Let’s construct a subset of $X$:

$Y=\{A\in{X}\:|\:A\notin{A}\}$.

Which proposition of the two is the true one $Y\in{Y}$ or $Y\notin{Y}$?

i) Let’s assume that the proposition $Y\in{Y}$ is the true one. In that case, since any element of $Y$ is a set that is not an element of itself, the proposition $Y\notin{Y}$ is true.

ii) Let’s assume that the proposition $Y\notin{Y}$ is the true one. In that case, according to the definition of $Y$, the proposition $Y\in{Y}$ is true. As a result, the following proposition has been proved:

# Set Theory

The concept of the “set” is one of the basic concepts of mathematics. In spite of this fact, there is no definition agreed on by the authority. Some mathematicians define sets as “the class of objects that have certain properties”. Although this definition is widespread, there are deficiencies.

Objects that form the set are called “elements”. The sets are represented with capital letters such as $A$, $B$, $C$, $X$, $Y$, and the elements of sets are represented with lower-case letter such as $a$, $b$, $c$, $x$, $y$. If $a$ is an element of $A$, this case is denoted by $a\in A$, if $a$ is not an element of $A$, this case is denoted by $a\notin A$. There are three types of representations to display the sets:

1. List method: In this representation, the elements of set are written into the curly braces, by putting the commas between the elements. In a set, an element cannot be written twice. As an example, $A=\{a,b,c,d,e\}$ can be given.

2. Venn diagram: In this representation, the elements of set are written inside a circle or rectangle. Let’s show the above example, $A=\{a,b,c,d,e\}$, by using Venn diagram: