1 What is Mathematics

Mathematics is a study of sets with structures. There are three basic structures.

• Algebraic Structure
• Topological Structure
• Ordered Structure
  
  For example if we are studying arithmetic of real numbers then we are studying the algebraic structure $<R,+,.>$. The study of Group theory is a study of the structure $<G,\ast >$, where $\ast$ is a binary operation on set $G$ satisfying certain properties. The study of metric spaces is the study of the structure $<X,d>$ where the function $d:X\times X\to R$ satisfies certain properties.
  
  Mathematics may broadly be categorised in the following branches.
• Pure Mathematics: Algebra(Group Theory, Ring Theory,Field Theory, Vector Spaces, Matrices and Determinants etc.), Analysis(Real Analysis, Complex Analysis), Topology, Differential Equations, Number Theory,Geometry...
• Discrete Mathematics: Combinatorics, Graph Theory, Lattice Theory,...
• Applied Mathematics: Differential Equations, Numerical Analysis, Operations Research, Cryptography, Computational Geometry, Statistics (as applied real analysis),...
  
  The above classification is not the precise one. The mode of study and approach of certain theory may categorise it in different branches.

2 Logic as foundation of Mathematics

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The Logic forms foundation for Mathematics. The Logic is the language of Mathematics. The rules of logic distinguishes between valid and invalid statements in Mathematics. Any proof in Mathematics can be translated into a sequence of logical statements and the validity of proofs can be established by using the rules of inferences. If there seems a problem with some proof (or argument), its validity may be checked by translating it into a sequence of logical statements and then applying rules of inferences.
The ideal course on Foundations of Mathematics will have the following outline.

Logic $\to$ Set theory $\to$ Number System

Ex. Is the statement $2\le 3$ true? Justify your answer.
Ans: The statement $2\le 3$ is logically equivalent to the statement $2<3\text{or}2=3$. Here the first statement is true and the second is false. The disjunction $p\vee q$ is true when $p$ is true and $q$ is false. Refer to the truth table of $p\vee q$. It now follows that the statement $2\le 3$ is true.

This example illustrates the use of rules of logic in understanding the clear meaning of statements in Mathematics. The or in Mathematics (disjunction $\vee$) is always inclusive or. The statement $p\vee q$ is true when at least one of $p$ and $q$ is true. The statement $p\vee q$ is true includes three possibilities.

• $p$ is true and $q$ is false
• $p$ is false and $q$ is true
• $p$ is true and $q$ is true

  It includes the third possibility. So is inclusive or. (There is an exclusive or in logic which is true when exactly one of $p$ and $q$ is true.)

Ex. Prove that the empty set $\varphi$ is subset of every set.
Ans: Let $S$ be a set. Consider the statement

It is a conditional statement $p\Rightarrow q$, where the statement $p$ is false and the statement $q$ may be true or false. If $p$ is false then the conditional statement $p\Rightarrow q$ is true (vacuously true). Refer to the truth table of $p\Rightarrow q$. It follows that the statement $\left(I\right)$ is true, which means that $\varphi$ is a subset of $S$.

Ex. Prove that the empty set $\varphi$ is a semi-group.
Ans: Semi-group: $G$ is a set and $\ast :G\times G$. The structure $<G,\ast >$ is a semi-group if

• $\ast$ is associative i.e. $\left(a\ast b\right)\ast c=$$a\ast \left(b\ast c\right)$ for all $a,b,c\in G$

The above definition can be written as
Semi-Group: $G$ is a set and $\ast :G\times G$. The structure $<G,\ast >$ is a semi-group if

• $a,b,c\in G$$\Rightarrow \left(a\ast b\right)\ast c=a\ast \left(b\ast c\right)$

The property $a,b,c\in G$$\Rightarrow \left(a\ast b\right)\ast c=$$a\ast \left(b\ast c\right)$ in a definition is a conditional statement $p\Rightarrow q$. For $G=\varphi$, the statement is $a,b,c\in \varphi$$\Rightarrow \left(a\ast b\right)\ast c=a\ast \left(b\ast c\right)$. The hypothesis $p:a,b,c\in \varphi$ is false. If $p$ is false then the conditional statement $p\Rightarrow q$ is true (vacuously true). Refer to the truth table of $p\Rightarrow q$. It follows that the statement $a,b,c\in \varphi$$\Rightarrow \left(a\ast b\right)\ast c=a\ast \left(b\ast c\right)$ is true, which means that $\varphi$ is a semi-group.

Ex. Prove that the empty set $\varphi$ is not a group.
Ans: In algebra texts, the reader may find the following two (equivalent) definitions of groups.
Group: $G$ is a non-empty set and $\ast :G\times G$. The structure $<G,\ast >$ is a group if

• $\ast$ is associative i.e. $\left(a\ast b\right)\ast c=$$a\ast \left(b\ast c\right)$ for all $a,b,c\in G$
• There is an identity element for $\ast$ in $G$ i.e. there is $e\in G$ such that $a\ast e=$$e\ast a=a$ for all $a\in G$
• Every element in $G$ has inverse with respect to $\ast$ i.e. for every $a\in G$ there is $b\in G$ such that $a\ast b=$$b\ast a=e$

OR
Group: $G$ is a set and $\ast :G\times G$. The structure $<G,\ast >$ is a group if

• $\ast$ is associative i.e. $\left(a\ast b\right)\ast c=$$a\ast \left(b\ast c\right)$ for all $a,b,c\in G$
• There is an identity element for $\ast$ in $G$ i.e. there is $e\in G$ such that $a\ast e=$$e\ast a=a$ for all $a\in G$
• Every element in $G$ has inverse with respect to $\ast$ i.e. for every $a\in G$ there is $b\in G$ such that $a\ast b=b\ast a=e$
  

According to the first definition of group, the empty set is not a group. The second definition can be written as
Group: $G$ is a set and $\ast :G\times G$. The structure $<G,\ast >$ is a group if

• $a,b,c\in G$$\Rightarrow \left(a\ast b\right)\ast c=a\ast \left(b\ast c\right)$
• $\exists e\in G$$\forall a\in G\left(a\ast e=e\ast a=a\right)$
• $\forall a\in G$$\exists b\in G\left(a\ast b=b\ast a=e\right)$

The second property is not true for the empty set as it doesn’t contain any element. Hence the empty set is not a group.

3 Role of definitions

There are two important properties of definitions.

• The definition is always fundamental in Mathematics.
  Let’s see an example.
  $i^2$
  $=i.i$
  $=\sqrt[]{-1}\sqrt[]{-1}$
  $=\sqrt[]{\left(-1\right)\left(-1\right)}$ $=\sqrt[]{1}$
  $=1$
  We proved that $i^2=1$! Where is the mistake? The mistake is in the step $\sqrt[]{a}\sqrt[]{b}=\sqrt[]{ab}$. This rule is valid only when $a$ and $b$ are non-negative. But more important question is What is the proper way to calculate the value of $i^2$. The answer is: use DEFINITION of product of two complex numbers. The complex numbers are defined as $a+ib$ where $a,b\in R$ and $i$ denotes the symbol $\sqrt[]{-1}$. Then addition, substraction, multiplication and division of complex numbers are defined. The multiplication of complex numbers is defined as
  
  $\left(a+ib\right)\left(c+id\right)$$=\left(ac-bd\right)+i\left(ad+bc\right)$ So the proper way to calculate the value of $i^2$ is
  $i^2$
  $=i.i$
  $=\left(0+1i\right)\left(0+1i\right)$
  $=\left(0-1\right)+\left(0+0\right)i$
  $=-1$

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  Definitions are always fundamental in Mathematics. Everything in Mathematics begins with definitions. We can’t argue with definitions. We have to accept definitions. If you think some argument is not correct,go back to the definitions and then start from it.

• Every definition in Mathematics is if and only if or biconditional ( $p\iff q)$ type of statement.
  Examples:
  Definition: A real number $x$ is positive if and only if $x>0$
  
  A real number $x$ is positive $\iff x>0$
  
  So if real number $x$ is positive then $x>0$. Conversely if $x>0$ then $x$ is positive real number.
  
  Definition: $△ABC$ is equilateral if and only if all sides of $△ABC$ have equal lengths.
  
  $△ABC$ is equilateral $\iff$ all sides of $△ABC$ have equal lengths.
  
  So if $△ABC$ is equilateral then all sides of $△ABC$ have equal lengths. Conversely if all sides of $△ABC$ have equal lengths then $△ABC$ is equilateral.
  
  It is important to note that Every definition in Mathematics is of this type. The standard convention is to write if instead of if and only if . So the above definitions could be stated as
  
  Definition: A real number $x$ is positive if $x>0$
  
  Definition: $△ABC$ is equilateral if all side of $△ABC$ have equal lengths.
  
  In above definitions, if means if and only if . The both way implications are used. For example, if $△ABC$ is equilateral then we shall use the fact that all side of $△ABC$ have equal lengths. On the other hand to prove that $△ABC$ is equilateral, we shall prove that all sides of $△ABC$ have equal lengths.

4 Axiomatic Theories in Modern Mathematics

The theories in Mathematics are of axiomatic types.

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There are defined/undefined concepts and axioms for these concepts. Based on these axioms, theories are developed. For example in Euclidean geometry, point is undefined concept and based on certain axioms; the Euclidean geometry is developed. In set theory, set is undefined concept and based on certain axioms; the set theory is developed. In many introductory texts on Real Analysis or Calculus, it is assumed that there is a set $R$ of real numbers (undefined concept) and algebraic,ordered and completeness properties (axioms) are stated based on which the properties of real numbers are proved.

5 Sets

The set is the most fundamental concept of Mathematics. The meaning is that anything and everything that we deal with in Mathematics is ultimately a set. For example, a relation (binary relation) on set $A$ is a subset (set) of $A\times A$. A function $f:A\to B$ is a subset (set) of $A\times B$. A sequence is a function and so it is also a set. The cardinal numbers are equivalence classes (of similarity relation) and so they are sets. The ordered pair $\left(x,y\right)$ is defined as $\left(x,y\right)=\left\{\left\{x\right\},\left\{x,y\right\}\right\}$ and so it is a set. The series of real numbers (defined as ordered pair of two sequences) is also a set. You may be surprised to know that numbers are also sets. For example,the natural numbers are defined as $0=\varphi ,1=\left\{0\right\}$$=\left\{\varphi \right\},2=\left\{1,0\right\}$$=\left\{\left\{\varphi \right\},\varphi \right\}\dots$.

The reader may find that the rigorous treatment of set theory is not so easy. It has taken centuries to formalize the logic and set theory in rigorous way. For example, what is the concept of the empty set? It is an axiom that the empty set exists. The empty set is a set containing no elements. Intuitively set can be thought as a collection of objects. So how a set can be empty? (The analogy may be helpful in this case. The sets can be thought as boxes containing objects and then the empty set is a box containing no object)

About the uniqueness of empty set. $A-A=\varphi$ and $B-B={\varphi }^{\text{'}}$ for sets $A$ and $B$. So whether both $\varphi$ and ${\varphi }^{\text{'}}$ are equal. It is by axiom that empty set is unique.

6 Cardinality of Sets

• Intuitively the cardinality of sets can be thought as a number of elements in a set. Two sets have same cardinality if there is one-to-one correspondence between them. If there is one to one correspondence between two sets then they are similar. The similarity is an equivalence relation on the collection of sets. The cardinality of a set may rigorously be defined as an equivalence class of this relation.
  
• A set is infinite if it is similar to a proper subset of itself.
• A set is finite if it is empty or can be put in one-to-one correspondence with the set $\left\{1,2,3,\dots ,n\right\}$ for some positive integer $n$.
• A set is denumerable if it can be put in one-to-one correspondence with the set $\left\{1,2,3,\dots ,\right\}$ for some positive integer $n$. The cardinality of denumerable sets is denoted by ${\chi }_0$.
• Empty set is the only set with cardinality $0$.
• Intuitively, a set is countable if its element can be counted as $1,2,3,\dots$ i.e. as first element, second element, third element,....A set is countable if it is finite or denumerable (Note: in some texts, denumerable sets are defined as coutable).
• A set is uncountable if it is not countable.

This proves that the set $I$ of positive integers is denumerable and hence counatable.



This proves that the set $2N$ of positive even integers is denumerable and hence countable.



This proves that the set $Z$ of integers is denumerable and hence countable.
The rationals too can be counted. So the set $Q$ of rationals is denumerable and hence countable.

• A subset of countable set is countable.
• A countable union of countable sets is countable. So a finite union of finite sets is finite(countable), a finite union of denumerable sets is denumerable(countable), a denumerable union of finite sets is countable and a denumerable union of denumerable sets is denumerable (countable).
• A finite cartesian product of finite sets is finite (countable).
• A finite cartesian product of denumerable sets is denumerable (countable).
• A denumerable cartesian product of denumerable sets is uncountable.
• Thus a countable cartesian product of countable sets may not be countable.
• There is no way of counting the elements of the set $Q^c$ of irrationals. Hence it is uncountable. If $a\ne b$ then intervals $\left(a,b\right).\left[a,b\right],\left[a,b\right),\left(a,b\right]$ are uncountable.
• The Cantors theorem says that the cardinality of a set $A$ is always less than the cardinality of its power set $P\left(A\right)$. So there is no largest set. For if $U$ is the largest set then $P\left(U\right)$contains more elements than $U$.
• ${\chi }_0$ denotes the cardinality of denumerable set. It is the smallest possible cardinal number that an infinite set can have.
• $\left|N\right|=\left|Z\right|$$=\left|Q\right|={\chi }_0$
• $\left|\left(a,b\right)\right|=$$\left|\left[a,b\right]\right|=$$\left|\left(a,b\right]\right|=\left|\left[a,b\right)\right|=$$\left|Q^{}\right|=$$\left|R\right|=$$2^{{\chi }_0}=c$
• $n<{\chi }_0<2^{{\chi }_0}$$<2^{2^{{\chi }_0}}$$<....$
  There are infinitely many infinites in cardinal numbers...the smallest (unity) is ${\chi }_0$.

7 Remarks on Definitions and Proofs

The definitions in Mathematics are of fundamental importance. It is advised to write definitions in few different (and equivalent) ways. Try to write every definition as a biconditional statement $p\iff q$. The clear understanding of definitions will ease out the learning curve of any theory. Once the definition is understood, you may try to write it in your own way. This may be used as a test of understanding.

The proofs are to be avoided in paragraphs. If proofs are written as a sequence of logical statements (step by step), they are better understood. This may not be possible for the lengthy and complex proofs. However proofs may be divided in parts and sub-proofs may be written as a sequence of logical statements. The proof is not completely understood unless one can view it as a single idea.

Most of the statements in Mathematics may be written as a conditional statement $p\Rightarrow q$. The direct proofs may be preferred whenever possible in proving conditional statements. In direct proofs, one begin with $p$ and after a sequence of logical statements it ends with $q$.

The one reason why Mathematics is difficult is because it is abstract. Whenever possible, graphs, pictures and diagrams should be drawn.Visualisations and analogies are helpful for understanding abstract concepts. This is specially true in calculus,analysis or metric spaces.