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## A Trip to Mathematics: Part III Relations and Functions

‘Michelle is the wife of Barak Obama.’
‘John is the brother of Nick.’
‘Robert is the father of Marry.’
‘Ram is older than Laxman.’
‘Mac is the product of Apple Inc.’
After reading these statements, you will realize that first ‘Noun’ of each sentence is some how related to other. We say that each one noun is in a RELATIONSHIP to other. Mischell is related to Barak Obama, as wife. John is related to Nick, as brother. Robert is related to Marry, as father. Ram is related to Laxman in terms of age(seniority). Mac is related to Apple Inc. as a product.These relations are also used in Mathematics, but a little variations; like ‘alphabets’ or ‘numbers’ are used at place of some noun and mathematical relations are used between them. Some good examples of relations are:

is less than
is greater than
is equal to
is an element of
belongs to
divides
etc. etc.

Some examples of regular mathematical statements which we encounter daily are:

4<6 : 4 is less than 6.
5=5 : 5 is equal to 5.
6/3 : 3 divides 6.

For a general use, we can represent a statement as:
”some x is related to y”
Here ‘is related to’ phrase is nothing but a particular mathematical relation. For mathematical convenience, we write ”x is related to y” as $x \rho y$. x and y are two objects in a certain order and they can also be used as ordered pairs (x,y).
$(x,y) \in \rho$ and $x \rho y$ are the same and will be treated as the same term in further readings. If $\rho$ represents the relation motherhood, then $\mathrm {(Jane, \ John)} \in \rho$ means that Jane is mother of John.
All the relations we discussed above, were in between two objects (x,y), thus they are called Binary Relations. $(x,y) \in \rho \Rightarrow \rho$ is a binary relation between a and b. Similarly, $(x,y,z) \in \rho \Rightarrow \rho$ is a ternary (3-nary) relation on ordered pair (x,y,z). In general a relation working on an n-tuple $(x_1, x_2, \ldots x_n) \in \rho \Rightarrow \rho$ is an n-ary relation working on n-tuple.
We shall now discuss Binary Relations more rigorously, since they have solid importance in process of defining functions and also in higher studies. In a binary relation, $(x,y) \in \rho$, the first object of the ordered pair is called the the domain of relation ρ and is defined by
$D_{\rho} := \{x| \mathrm{for \ some \ y, \ (x,y) \in \rho} \}$ and also the second object is called the range of the relation ρ and is defined by $R_{\rho} := \{y| \mathrm{for \ some \ y, \ (x,y) \in \rho} \}$.
There is one more thing to discuss about relations and that is about equivalence relation.
A relation is equivalence if it satisfies three properties, Symmetric, Reflexive and Transitive.
I mean to say that if a relation is symmetric, reflexive and transitive then the relation is equivalence. You might be thinking that what these terms (symmetric, reflexive and transitive) mean here. Let me explain them separately:
A relation is symmetric: Consider three sentences “Jen is the mother of John.”; “John is brother of Nick.” and “Jen, John and Nick live in a room altogether.”
In first sentence Jen has a relationship of motherhood to John. But can John have the same relation to Jen? Can John be mother of Jen? The answer is obviously NO! This type of relations are not symmetric. Now consider second statement. John has a brotherhood relationship with Nick. But can Nick have the same relation to John? Can Nick be brother of John? The answer is simply YES! Thus, both the sentences “John is the brother of Nick.” and “Nick is the brother of John.” are the same. We may say that both are symmetric sentences. And here the relation of ‘brotherhood’ is symmetric in nature. Again LIVING WITH is also symmetric (it’s your take to understand how?).
Now let we try to write above short discussion in general and mathematical forms. Let X and Y be two objects (numbers or people or any living or non-living thing) and have a relation ρ between them. Then we write that X is related by a relation ρ , to Y. Or X ρ Y.
And if ρ is a symmetric relation, we might say that Y is (also) related by a relation ρ to X. Or Y ρ X.
So, in one line; $X \rho Y \iff Y \rho X$ is true.

A relation is reflexive if X is related to itself by a relation. i.e., $X \rho X$. Consider the statement “Jen, John and Nick live in a house altogether.” once again. Is the relation of living reflexive? How to check? Ask like, Jen lives with Jen, true? Yes! Jen lives there.
A relation is transitive, means that some objects X, Y, Z are such that if X is related to Y by the relation, Y is related to Z by the relation, then X is also related to Z by the same relation.
i.e., $X \rho Y \wedge Y \rho Z \Rightarrow X \rho Z$. For example, the relationship of brotherhood is transitive. (Why?) Now we are able to define the equivalence relation.
We say that a relation ρ is an equivalence relation if following properties are satisfied: (i) $X \rho Y \iff Y \rho X$
(ii) $X \rho X$
(iii) $X \rho Y \ Y \rho Z \Rightarrow X \rho Z$.

Functions: Let f be a relation (we are using f at the place of earlier used ρ ) on an ordered pair $(x,y) : x \in X \ y \in Y$. We can write xfy, a relation. This relation is called a function if and only if for every x, there is always a single value of y. I mean to say that if $xfy_1$ is true and $xfy_2$ is also true, then always $y_1=y_2$. This definition is standard but there are some drawbacks of this definition, which we shall discuss in the beginning of Real Analysis .
Many synonyms for the word ‘function’ are used at various stages of mathematics, e.g. Transformation, Map or Mapping, Operator, Correspondence. As already said, in ordered pair (x,y), x is called the element of domain of the function (and X the domain of the function) and y is called the element in range or co-domain of the function (and Y the range of the function).

Here I will stop myself. I don’t want a post to be long (specially when writing on basic mathematics) that reader feel it boring. The intermediate mathematics of functions is planned to be discussed in Calculus and advanced part in functional analysis. Please note that I am regularly revising older articles and trying to maintain the accuracy and completeness. If you feel that there is any fault or incompleteness in a post then please make a comment on respective post. If you are interested in writing a guest article on this blog, then kindly email me at mdnb[at]live[dot]in.

# Introduction

In English dictionary, the word Set has various meanings. It is often said to be the word with maximum meanings (synonyms). But out of all, we should consider only one meaning: ”collection of objects” — a phrase that provides you enough clarity about what Set is all about. But It is not the exact mathematical definition of Set . The theory of Set as a mathematical discipline rose up with George Cantor, German mathematician. It is said that Cantor was working on some problems in Trigonometric series and series of real numbers, which accidently led him to recognise the importance of some distinct collections and intervals. And he started developing Set Theory. Well, we are not here to discuss the history of sets; but Mathematical importance.

Cantor defined the set as a ‘plurality concieved as a unity’ (many in one; in other words, mentally putting together a number of things and assigning them into one box).
Mathematically, a Set $S$ is ‘any collection’ of definite, distinguishable objects of our universe, concieved as a whole. The objects (or things) are called the elements or members of the set $S$. Some sets which are often termed in real life are, words like ”bunch”, ”herd”, ”flock” etc. The set is a different entity from any of its members.
For example, a flock of birds (set) is not just only a single bird (member of the set). ‘Flock’ is just a concept with no material existence but ‘Bird’ or ‘birds’ are real.

# Representing sets

Sets are represented in two main ways:
1. Standard Method: In this method we use to write all elements of a set in a curly bracket ( { } ).
For example:
Flock of Birds := {Bird-1, Bird-2, …, Bird-100,…}
or, $F:= \{B_1, B_2, \ldots, B_{100}, \ldots \}$ is a set.
Here I have used first capital letter of each term to notate the example mathematically. We read this set as, A set F is defined by a collection of objects $B_1, B_2, \ldots$ etc.
2. Characteristic Method: In this method, we write a representative element and define that by a characteristic property. A characteristic property of a set is a property which is satisfied by each member of that set and by nothing else.
For example, above set of Flock of birds can also be written as:
$\mathrm{ F := \{ B : B \ is \ a \ bird \} }$
which has the same meaning at a wider extent. We read it as: ”A set F is defined by element B such that B is a bird.”

# Standard Sets

Some standard sets in Mathematics are:

Set of Natural Numbers: It includes of the numbers, which we can count, viz. $\mathrm{ \{0,1,2,3,4,5,6,7, \ldots \}}$. The set of natural numbers is denoted by $\mathbb{N}$.

Set of Integers: Integers includes of negatives of natural numbers and natural numbers itself. It is denoted by $\mathbb{Z}$. $-5, -4, 1, 2, 0$ …all are integers. The rigorous definition of integers be discussed in fourth part of the series.

Set of Rational Numbers: Rational numbers are numbers which might be represented as $\frac{p}{q}$, where p and q both are integers and relatively prime to each other and q not being zero. The set of rational numbers is represented by $\mathbb{Q}$ and may include elements like $\frac{2}{3}, \frac{-5}{7}, 8$. The characteristic notation of the set of rational numbers is $\mathbb{Q} := \{ \dfrac{p}{q};/ p,q \in \mathbb{Z}, \ (p,q) \equiv 1 \ q \ne 0 \}$. The rigorous dicussion about rational numbers will be provided in fourth part of the series.

Empty Set: It is possible to conceive a set with no elements at all. Such a set is variously known as an empty set or a void set or a vacuous set or a null set.
An example of emptyset is the set $\{\mathrm{x:\ x \ is \ an \ integer \ and \ x^2=2} \}$, since there exists no integer which square is 2 —the set is empty. The unique empty set is denoted by $\emptyset$.
Unit Set: A set with only one element is called the unit set. {x} is a unit set.

Universal Set: A set which contains every possible element in the universe, is a universal set. It is denoted by $U$.

# Two Sets

Let $A$ and $B$ be two sets. We say that $A$ is a subset of $B$ (or $B$ is superset of $A$ or $A$ is contained in $B$ or $B$ contains $A$) if every element of $A$ is also an element of set $B$. In this case we write, $A \subseteq B$ or $B \supseteq A$ respectively, having the same meaning .
Two sets are equal to each other if and only if each is a subset of the other. Subset word might be understood using ‘sub-collection’ or ‘subfamily’ as its synonyms.
Operations on Sets:
As Addition, Subtraction, Multiplication and Division are the most common mathematical operations between numbers; Union, Intersection, Complement, Symmetric difference, Cartesian Products are the same between sets.

UNION OF SETS

If A and B are two sets, then their union (or join) is the set, defined by another set $S \cup T$ such that it consists of elements from either A or B or both. If we write the sets A and B using Characteristic Method as,

$\mathrm{A:= \{x : x \ is \ an \ element \ of \ set \ A\}}$.
and,$\mathrm{B:=\{x : x \ is \ an \ element \ of \ set \ B\}}$
then the union set of A and B is defined by set J such that

$\mathrm{J := \{x: x \ is \ an \ element \ of \ set \ A \ or \ set \ B \ or \ both \}}$.

For practical example, let we have two sets:
$A:= \{1,2,3,r,t,y\}$ and $B:=\{3,6,9,r,y,g,k\}$ be any two sets; then their union is $A \cup B :=\{ 1,2,3,6,9,r,t,y,g,k \}$.
Note that it behaves like writting all the elements of each set, just caring that you are not allowed to write one element twice.

Here is a short video explaining Unions of Sets:

INTERSECTION OF SETS

Intersection or meet of two sets A and B is similarly defined by ‘and’ connective. The set {x: x is an element of A and x is an element of B} or briefly $\mathrm { \{ x: x \in A \wedge x \in B \}}$. It is denoted by $A \cap B$ or by $A \cdot B$ or by $AB$.
For example, and by definition, if A and B be two sets defined as,
$A:=\{1,2,3,r,t,y\}$
$B:=\{3,6,9,r,g,k\}$
then their intersection set, defined by $A \cap B:= \{3,r\}$.

In simple words, the set formed with all common elements of two or more sets is called the intersection set of those sets.
Here is a video explaining the intersection of sets:

If, again, A and B are two sets, we say that A is disjoint from B or B is disjoint from A or both A and B are mutually disjoint, if they have no common elements. Mathematically, two sets A and B are said to be disjoint iff $A \cap B := \emptyset$ .
If two sets are not disjoint, they are said to intersect each other.

PARTITION OF A SET

A partition set of a set X is a disjoint collection of non-empty and distinct subsets of X such that each member of X is a member of exactly one member (subset) of the collection.
For example, if $\{q,w,e,r,t,y,u\}$ is a set of keyboard letters, then $\{ \{q,w,e\}, \{r\}, \{t,y\},\{u\}\}$ is a partition of the set and each element of the set belongs to exactly one member (subset) of partition set. Note that there are many partition sets possible for a set. For example, $\{\{q,w\}, \{e,r\},\{t,y,u\}\}$ is also a partition set of set $\{q,w,e,r,t,y,u\}$.
A Video on Partition of set:

COMPLEMENT SET OF A SET

The complement set $A^c$ of a set $A$ is a collection of objects which do not belong to $A$. Mathematically, $A^c := \{x: x \notin A \}$.

The relative complement of set $A$ with respect to another set $X$ is $X \cap A^c$ ; i.e., intersection of set $X$ and the complement set of $A$. This is usually shortened by $X-A$, read X minus A. Thus, $X-A := {x : x \in X \wedge x \notin A}$, that is, the set of members of $X$ which are not members of $A$.

The complement set is considered as a relatative complement set with respect to (w.r.t) the universal set, and is called the Absolute Complement Set.

A Video on Complement  of  A Set:

SYMMETRIC DIFFERENCE

The symmetric difference is another difference of sets $A$ and $B$, symbolized $A \Delta B$, is defined by the union of mutual complements of sets $A$ and $B$, i.e., $A \Delta B := (A -B) \cup (B-A) = B \Delta A$.

# Theorems on Sets

1. $A \cup (B \cup C) = (A \cup B) \cup C$
2. $A \cap (B \cap C) = (A \cap B) \cap C$
3. $A \cup B= B \cup A$
4. $A \cap B= B \cap A$
5. $A \cup (B \cap C)= (A \cup B) \cap (A \cup C)$
6. $A \cap (B \cup C)= (A \cap B) \cup (A \cap C)$
7. $A \cup \emptyset= A$
8. $A \cap \emptyset= \emptyset$
9. $A \cup U=U$
10. $A \cap U=A$
11. $A \cup A^c=U$
12. $A \cap A^c=\emptyset$
13. If $\forall A \ , A \cup B=A$ $\Rightarrow B=\emptyset$
14. If $\forall A \ , A \cap B=A \Rightarrow B=U$
15. Self-dual Property: If $A \cup B =U$ and $A \cap B=\emptyset \ \Rightarrow B=A^c$
16. Self Dual: ${(A^c)}^c=A$
17. ${\emptyset}^c=U$
18. $U^c= \emptyset$
19. Idempotent Law: $A \cup A=A$
20. Idempotent Law: $A \cap A =A$
21. Absorption Law: $A \cup (A \cap B) =A$
22. Absorption Law: $A \cap (A \cup B) =A$
23. de Morgen Law: ${(A \cup B)}^c =A^c \cap B^c$
24. de Morgen Law: ${(A \cap B)}^c =A^c \cup B^c$

# Another Theorem

The following statements about set A and set B are equivalent to one another

1. $A \subseteq B$
2. $A \cap B=A$
3. $A \cup B =B$

I trust that we are familiar with the basic properties of complements, unions and intersections. We should now turn to another very important concept, that of a function. So how to define a function? Have we any hint that can lead us to define one of the most important terms in mathematics? We have notion of Sets. We will use it in an ordered manner, saying that an ordered pair.
First of all we need to explain the the notion of an ordered pair. If $x$ and $y$ are some objects, how should we define the ordered pair $(x,y)$ of those objects? By another set? Yes!! The ordered pair is also termed as an ordered set. We define ordered pair $(x,y)$ to be the set $\{\{x,y\},\{x\}\}$. We can denote the ordered pair $(x,y)$ by too, if there is a desperate need to use the small bracket ‘( )’ elsewhere.
So, note that Ordered Pairs
$(x,y) := \{\{x,y\},\{x\}\}$
and $(y,x) :=\{\{y,x\},\{y\}\} =\{\{x,y\},\{y\}\}$ are not identical. Both are different sets.
You might think that if ordered pair can be defined with two objects, then why not with three or more objects. As we defined ordered pair (ordered double, as a term) $(x,y)$, we can also define $(x,y,z)$, an ordered triple. And similarly an ordered $n$ -tuple $(x_1, x_2, \ldots x_n)$ in general such that:
$(x,y):=\{\{x,y\},\{x\}\}$
$(x,y,z):=\{\{x,y,z\},\{x,y\},\{x\}\}$
$(x_1, x_2, x_3, \ldots x_n) := \{\{x_1, x_2, x_3, \ldots x_n\}, \{x_1, x_2, x_3, \ldots x_{n-1}\}, \ldots, \{x_1, x_2\}, \{x_1\}\}$.
Another important topic, which is very important in process to define function (actually in process to define ordered pair) is Cartesian Product (say it, Product, simply) of two sets. Let $A$ and $B$ be two sets. Then their Product (I said, we’ll not use Cartesian anymore) is defined to be the (another) set of an ordered pair, $(a,b)$, where $a$ and $b$ are the elements of set $A$ and set $B$ respectively. Mathematically; the product of two sets $A$ and $B$
$A \times B := \{(a,b) : a \in A, \ b \in B\}$.
Note that $A \times B \ne B \times A$.
The name as well as the notation is suggestive in that if $A$ has $m$ elements, $B$ has $n$ elements then $A \times B$ indeed has $mn$ elements.
We see that if we product two sets, we get an ordered pair of two objects (now we’ll say them, variables). Similarly if we product more than two sets we get ordered pair of same number of variables. For example:
$X \times Y := (x,y); x \in X, y \in Y$.
$X \times Y \times Z :=(x,y,z); x \in X, y \in Y, z \in Z$. etc.
The sets, which are being product are called the factor sets of the ordered pair obtained. When we form products, it is not necessary that the factor sets be distinct. The product of the same set $A$ taken $n$ times is called the $n$ -th power of $A$ and is denoted by $A^n$. Thus, $A^2$ is $A \times A$. $A^3$ is $A \times A \times A$. And so on.

Now we are ready to define functions. The next part of this series will focus on functions.

# About “A Trip To Mathematics”:

A Trip to Mathematics is an indefinitely long series, aimed on generally interested readers and other undergraduate students. This series will deal Basic Mathematics as well as Advanced Mathematics in very interactive manners. Each post of this series is kept small that reader be able to grasp concepts. Critics and suggestions are invited in form of comments.

# What is Logic?

If mathematics is regarded as a language, then logic is its grammar.
In other words, logical precision has the same importance in mathematics as grammatical accuracy in a language. As linguistic grammar has sentences, statements— logic has them too. Let we discuss about Sentence &Statements, then we shall proceed to further logic .

# Sentence & Statements

A sentence is a collection of some words, those together having some sense.

For example:

1. Math is a tough subject.
2. English is not a tough subject.
3. Math and English both are tough subjects.
4. Either Math or English is tough subject.
5. If Math is a tough subject, then English is also a tough subject.
6. Math is a tough subject, if and only if English is a tough subject.

Just have a quick look on above collections of words. Those are sentences, since they have some meaning too. First sentence is called Prime Sentence, i.e., sentence which either contains no connectives or, by choice, is regarded as “indivisible”. The five words

• not
• and
• or
• if …. then
• if and only if

or their combinations are called ‘connectives‘. The sentences (all but first) are called composite sentences, i.e., a declarative sentence (statement) in which one or more connectives appear. Remember that there is no difference between a sentence and statement in general logic. In this series, sentences and statements would have the same meaning.

# Connectives

not: A sentence which is modified by the word “not” is called the negation of the original sentence.
For example: “English is not a tough subject” is the negation of “English is a tough subject“. Also, “3 is not a prime” is the negation of “3 is a prime“. Always note that negation doesn’t really mean the converse of a sentence. For example, you can not write “English is a simple subject” as the negation of “English is a tough subject“.
In mathematical writings, symbols are often used for conciseness. The negation of sentences/statements is expressed by putting a slash (/) over that symbol which incorporates the principal verb in the statement.
For example: The statement $x=y$ (read ‘x is equal to y’) is negated as $x \ne y$ (read ‘x is not equal to y‘). Similarly, $x \notin A$ (read ‘x does not belong to set A‘) is the negation of $x \in A$ (read ‘x belongs to set A‘).
Statements are sometimes represented by symbols like p, q, r, s etc. With this notation there is a symbol, $\not$ or ¬ (read as ‘not’) for negation. For example if ‘p’ stands for the statement “Terence Tao is a professor” then $\not p$ [or ¬p] is read as ‘not p’ and states for “Terence Tao is not a professor.” Sometimes ~p is also used for the negation of p.
and: The word “and” is used to join two sentences to form a composite sentence which is called the conjunction of the two sentences. For example, the sentence “I am writing, and my sister is reading” is the conjunction of the two sentences: “I am writing” and “My sister is reading“. In ordinary language (English), words like “but, while” are used as approximate synonyms for “and“, however in math, we shall ignore possible differences in shades of meaning which might accompany the use of one in the place of the other. This allows us to write “I am writing but my sister is reading” having the same mathematical meaning as above.
The standard notation for conjunction is $\wedge$, read as ‘and‘. If p and q are statements then their conjunction is denoted by $p \wedge q$ and is read as ‘p and q’.
or: A sentence formed by connecting two sentences with the word “or” is called the disjunction of the two sentences. For example, “Justin Bieber is a celebrity, or Sachin Tendulkar is a footballer.” is a disjunction of “Justin Bieber is a celebrity” and “Sachin Tendulkar is a footballer“.
Sometimes we put the word ‘either‘ before the first statement to make the disjunction sound nice, but it is not necessary to do so, so far as a logician is concerned. The symbolic notation for disjunction is $\vee$ read ‘or’. If p and q are two statements, their disjunction is represented by $p \vee q$ and read as p or q.

if….then: From two sentences we may construct one of the from “If . . . . . then . . .“; which is called a conditional sentence. The sentence immediately following IF is the antecedent, and the sentence immediately following THEN is the consequent. For example, “If 5 <6, 6<7, then 5<7” is a conditional sentence whith “5<6, 6<7” as antecedent and “5<7” as consequent. If p and q are antecedent and consequent sentences respectively, then the conditional sentence can be written as:

“If p then q”.

This can be mathematically represented as $p \Rightarrow q$ and is read as “p implies q” and the statement sometimes is also called implication statement. Several other ways are available to paraphrase implication statements including:

1. If p then q
2. p implies q
3. q follows from p
4. q is a logical consequence of p
5. p (is true) only if q (is true)
6. p is a sufficient condition for q
7. q is a necessary condition for p

If and Only If:  The phrase “if and only if” (abbreviated as ‘iff‘) is used to obtain a bi-conditional sentence. For example, “A triangle is called a right angled triangle, if and only if one of its angles is 90°.” This sentence can be understood in either ways: “A triangle is called a right angled triangle if one of its angles is 90°” and “One of angles of a triangle is 90° if the triangle is right angled triangle.” This means that first prime sentence implies second prime sentence and second prime sentence implies first one. (This is why ‘iff’ is sometimes called double-implication.)

Another example is “A glass is half filled iff that glass is half empty.

If p and q are two statements, then we regard the biconditional statement as “p if and only if q” or “p iff q” and mathematically represent by ” $p\iff q$ “. $\iff$ represents double implication and read as ‘if and only if’.

In the statement $p \iff q$, the implication $p \Rightarrow q$ is called direct implication and the implication $q \Rightarrow p$ is called the converse implication of the statement.

# Other terms in logic:

Stronger and Weaker Statements: A statement p is stronger than a statement q (or that q is weaker than p) if the implication statement $p \Rightarrow q$ is true.
Strictly Stronger and Strictly Weaker Statements: The word ‘stronger‘(or weaker) does not necessarily mean ‘strictly stronger‘ (or strictly weaker).
For example, every statement is stronger than itself, since $p \Rightarrow p$. The apparent paradox here is purely linguistical. If we want to avoid it, we should replace the word stronger by the phrase ‘stronger than‘ or ‘possibly as strong as‘.

If $p \Rightarrow q$ is true but its converse is false ($q \not \Rightarrow p$), then we say that p is strictly stronger than q (or that q is strictly weaker than p). For example it is easy to say that a given quadrilateral is a rhombus that to say it is a parallelogram. Another understandable example is that ” If a blog is hosted on WordPress.com, it is powered with WordPress software.” is true but ” If a blog is powered with WordPress software , it is hosted on WordPress.com” is not true.

# Logical Approach

What exactly is the difference between a mathematician, a physicist and a layman?
Let us suppose they all start measuring the angles of hundreds of triangles of various shapes, find the sum in each case and keep a record.

Suppose the layman finds that with one or two exceptions the sum in each case comes out to be 180 degrees. He will ignore the exceptions and state ‘The sum of the three angles in a triangle is 180 degrees.’

A physicist will be more cautious in dealing the exceptional cases. He will examine then more carefully. If he finds that the sum in them some where 179 degrees to 181 degrees, say, then if will attribute the deviation to experimental errors. He will state a law – ‘The sum of the three angles of any triangle is 180 degrees.’ He will then watch happily as the rest of the world puts his law to test and finds that it holds good in thousands of different cases, until somebody comes up with a triangle in which the law fails miserably. The physicist now has to withdraw his law altogether or else to replace it by some other law which holds good in all the cases tried. Even this new law may have to be modified at a later date. And this will continue without end.

A mathematician will be the fussiest of all. If there is even a single exception, he will refrain from saying anything. Even when millions of triangles are tried without a single exception, he will not state it as a theorem that the sum of the three angles in ‘any’ triangle is 180 degrees. The reason is that there are infinitely many different types of triangles. To generalise from a million to infinity is as baseless to a mathematician as to generalise from one to a million. He will at the most make a conjecture and say that there is a ‘strong evidence’ suggesting that the conjecture is true.

The approach taken by the layman or the physicist is known as the inductive approach whereas the mathematician’s approach is called the deductive approach.

# Inductive Approach

In inductive approach, we make a few observations and generalise. Exceptions are generally not counted in inductive approach.

# Deductive Approach

In this approach, we deduce from something which is already proven.

# Axioms or Postulates

Sometimes, when deducting theorems or conclusion from another theorems, we reach at a stage where a certain statement cannot be proved from any ‘other’ proved statement and must be taken for granted to be true, then such a statement is called an axiom or a postulate.
Each branch of mathematics has its own populates or axioms. For example, the most fundamental axiom of geometry is that infinitely many lines can be drawn passing through a single point. The whole beautiful structure of geometry is based on five or six such axioms and every theorem in geometry can be ultimately Deducted from these axioms.

# Argument, Premises and Conclusion

An argument is really speaking nothing more than an implication statement. Its hypothesis consists of the conjunction of several statements, called premises. In giving an argument, its premises are first listed (in any order), then connecting all, a conclusion is given. Example of an argument:
Premises:   $p_1$         Every man is mortal.
$p_2$                              Ram is a man.
———————————————————————————-
Conclusion:                   $q$ Ram is mortal.

Symbolically, let us denote the premises of an argument by $p_1, p_2, \ldots , p_n$ and its conclusion by $q$. Then the argument is the statement $(p_1 \wedge p_2 \wedge \ldots \wedge p_n) \Rightarrow q$. If this implication is true, the argument is valid otherwise it is invalid.

To be continued……

## The problem of the Hundred Fowls

This is a popular Chinese problem, on Linear Diophantine equations, which in wording seems as a puzzle or riddle. However, when used algebraic notations, it looks obvious. The problems states :

 If a cock is worth 5 coins, a hen 3 coins, and three chickens together 1 coin, how many cocks, hens and chickens, totaling 100 in number, can be bought for 100 coins?

This puzzle in terms of algebraic equations can be written as $5x+3y+\frac{1}{3}z=100$ and $x+y+z=100$
where $x, y, z$ being the number of cocks, hens and chicks respectively.
We find that there are two equations with three unknown quantities. So eliminating one of the unknowns, by putting $z=100-x-y$ from second equation into first one such that $5x+3y+\frac{1}{3} (100-x-y)=100$
or, $15x+9y+100-x-y=300$
or, $14x+8y=200$
or, $7x+4y=100$.
Which is a linear Diophantine equation (with only two unknown quantities).
The equation $7x+4y=100$ has the general solution   [links to WolframAlpha] $x=4 t$ and $y=25-7t$, so that $z=75+3t$ where $t$ is an arbitrary integer.
Now, since $x, y, z$ are the number of creatures, hence $x, y, z >0$ and thus $4t >0$ , $25-7t >0$ and $75+3t >0$ which imply that $0 < t < 3\frac{4}{7}$. And because t must have integer values, we have $t=1,2,3$. Which gives the following three solutions:

 Values of $t$ No. Of cocks ( $x=4 t$ ) No. Of hens ($y=25-7t$) No. Of chicks ($z=75+3t$) 1 4 18 78 2 8 11 81 3 12 4 84

So there are the three ways to chose the number of cocks, hens and chicken totaling 100 to buy for 100 coins.

Problem Sources:
Elementary Number Theory
David M. Burton, 2006
McGrawHill Publications

Wikipedia article on Diophantine Equations

Image Credit

## Fermat Numbers

Fermat Number, a class of numbers, is an integer of the form $F_n=2^{2^n} +1 \ \ n \ge 0$.

For example: Putting $n := 0,1,2 \ldots$ in $F_n=2^{2^n}$ we get $F_0=3$, $F_1=5$, $F_2=17$, $F_3=257$ etc.

Fermat observed that all the integers $F_0, F_1, F_2, F_3, \ldots$ were prime numbers and announced that $F_n$ is a prime for each natural value of $n$.

In writing to Prof. Mersenne, Fermat confidently announced:

I have found that numbers of the form $2^{2^n}+1$ are always prime numbers and have long since signified to analysts the truth of this theorem.

However, he also accepted that he was unable to prove it theoretically. Euler in 1732 negated Fermat’s fact and told that $F_1 -F_4$ are primes but $F_5=2^{2^5} =4294967297$ is not a prime since it is divisible by 641.
Euler also stated that all Fermat numbers are not necessarily primes and the Fermat number which is a prime, might be called a Fermat Prime. Euler used division to prove the fact that $F_5$ is not a prime. The elementary proof of Euler’s negation is due to G. Bennett.

# Theorem:

The Fermat number $F_5$ is divisible by $641$ i.e., $641|F_5$.

# Proof:

As defined $F_5 :=2^{2^5}+1=2^{32}+1 \ \ldots (1)$

Factorising $641$ in such a way that $641=640+1 =5 \times 128+1 \\ =5 \times 2^7 +1$
Assuming $a=5 \bigwedge b=2^7$ we have $ab+1=641$.

Subtracting $a^4=5^4=625$ from 641, we get $ab+1-a^4=641-625=16=2^4 \ \ldots (2)$.

Now again, equation (1) could be written as
$F_5=2^{32}+1 \\ \ =2^4 \times {(2^7)}^4+1 \\ \ =2^4 b^4 +1 \\ \ =(1+ab-a^4)b^4 +1 \\ \ =(1+ab)[a^4+(1-ab)(1+a^2b^2)] \\ \ =641 \times \mathrm{an \, Integer}$
Which gives that $641|F_n$.

Mathematics is on its progression and well developed now but it is yet not confirmed that whether there are infinitely many Fermat primes or, for that matter, whether there is at least one Fermat prime beyond $F_4$. The best guess is that all Fermat numbers $F_n>F_4$ are composite (non-prime).
A useful property of Fermat numbers is that they are relatively prime to each other; i.e., for Fermat numbers $F_n, F_m \ m > n \ge 0$, $\mathrm{gcd}(F_m, F_n) =1$.

Following two theorems are very useful in determining the primality of Fermat numbers:

# Pepin Test:

For $n \ge 1$, the Fermat number $F_n$ is prime $\iff 3^{(F_n-1)/2} \equiv -1 \pmod {F_n}$

# Euler- Lucas Theorem

Any prime divisor $p$ of $F_n$, where $n \ge 2$, is of form $p=k \cdot 2^{n+2}+1$.

Fermat numbers ($F_n$) with $n=0, 1, 2, 3, 4$ are prime; with $n=5,6,7,8,9,10,11$ have completely been factored; with $n=12, 13, 15, 16, 18, 19, 25, 27, 30$ have two or more prime factors known; with $n=17, 21, 23, 26, 28, 29, 31, 32$ have only one prime factor known; with $n=14,20,22,24$ have no factors known but proved composites. $F_{33}$ has not yet been proved either prime or composite.

## How to Draw the Famous Batman Curve

The ellipse $\displaystyle \left( \frac{x}{7} \right)^{2} + \left( \frac{y}{3} \right)^{2} - 1 = 0$ looks like this:

So the curve $\left( \frac{x}{7} \right)^{2}\sqrt{\frac{\left| \left| x \right|-3 \right|}{\left| x \right|-3}} + \left( \frac{y}{3} \right)^{2}\sqrt{\frac{\left| y+3\frac{\sqrt{33}}{7} \right|}{y+3\frac{\sqrt{33}}{7}}} - 1 = 0$ is the above ellipse, in the region where $|x|>3$ and $y > -3\sqrt{33}/7$:

That’s the first factor.
(more…)

## Pi (Π)

Pi, the mathematical constant, has a yet to determine value. As of now following sequence of digits has found to be most accurate for the approximation for $\pi$. :3.1415926535897932384626433832795028841971693993751058209 7494459230781640628620899862803482534211706798214808651 3282306647093844609550582231725359408128481117450284102 7019385211055596446229489549303819644288109756659334461 2847564823378678316527120190914564856692346034861045432 6648213393607260249141273724587006606315588174881520920 9628292540917153643678925903600113305305488204665213841 4695194151160943305727036575959195309218611738193261179 3105118548074462379962749567351885752724891227938183011 9491298336733624406566430860213949463952247371907021798 6094370277053921717629317675238467481846766940513200056 8127145263560827785771342757789609173637178721468440901 2249534301465495853710507922796892589235420199561121290 2196086403441815981362977477130996051870721134999999837 2978049951059731732816096318595024459455346908302642522 3082533446850352619311881710100031378387528865875332083 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