ZENO'S DICHOTOMY PARADOX | ANCIENT MATH | MYTHOLOGY OF MATH

INTRODUCTION:

I am damn sure that you never heard this name before.If you know this before then you are special.

     Don't get sad we will discuss about it briefly and from basic.

ZENO:

Zeno of Elea was a pre-Socratic Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes.

What is Paradox?

A paradox is a statement that contradicts itself, or that must be both true and untrue at the same time. ... But a key part of paradoxes is that they at least sound reasonable.

Dichotomy Paradox:

Dichotomy Paradox means before before an object can travel a given distance , it must travel a distance . In order to travel , it must travel , etc. Since this sequence goes on forever, it therefore appears that the distance cannot be traveled.

Simply,

In Dichotomy Paradox, Zeno argued that a runner will never reach the stationary goal line on a straight racetrack. The reason is that the runner must first reach half the distance to the goal, but when there he must still cross half the remaining distance to the goal, but having done that the runner must cover half of the new remainder, and so on.

Example-1:  If the goal is one meter away, the runner must cover a distance of 1/2 meter, then 1/4 meter, then 1/8 meter, and so on ad infinitum.So the runner cannot reach the final goal.

EXAMPLE-2: you may know about the story of Rabbit and Tortoise Race.

According to zeno's paradox if the Rabbit start the race after tortoise start then rabbit will not overtake the tortoise.

Don't Confuse with this because Dichotomy Paradox means The Contradiction of itself which it true and false at same time.

   I know this is funny but it is real.

Every time the hare reaches the place where the tortoise was, the tortoise has moved on a bit further. (Image)

And if the Pictures continue to infinity then you will find that Tortoise  will be bit ahead of Rabbit with less than a nanometer or picometer of whatever..

So Now the question must arises that

1. what is the time takes to reach at the goal?

2. In real life we reach at a goal in finite time , So in Zeno,s Paradox is it finite or infinite?

3. Is Zeno's Paradox is just a illusion or works in real life?

Time Management of Zeno's Dichotomy Paradox:

Let

      Distance=2KM

      Velocity = 1km/hr

From Zeno's Point of view

• for Covering 1st half i.e 1km will take 1hour
• For covering 2nd portion i.e; 500m will take 1/2 hr.
• For Covering 3rd portion i.e; 250m will take 1/4 hr.
• And So On.....

So Total Time will be (in Hours)

   = 1+1/2+1/4+1/8+1/16+........

   = 1+2-1 +2-2 +2-3 +....

   It is a Geometric Progression

Here a=1 and r=2-1

It is a Infinite series so 1/(1-r) will be the Formula

=1/1-2-1

=2  ( Proof)

Now you can see that the answer is correct according to real life.

RANDOM VARIABLE

INTRODUCTION:

Some Thinks that Probability is very easy 

And Some think it is very tough.

      If you Think Probability is tough then Proceed to our Probability blogs to learn easily.

Let's Proceed to our Topic

✦ A random variable is often described as a variable whose values are determines  by outcomes of a random experiment.

OR, 

      We assigned some real values to each of the outcomes of a sample space ,which is called the random variable.

✦ A random variable is a function , X :S ➝ R i.e a random variable is a function whose domain is the sample space of a random experiment(S) and whose range is a real line(R).

 Example: Let X is a random variable , the no. of heads in the experiment of tossing a coin twice , then in this case  S ={ HH,HT,TH,TT} and X ={2,1,1,0} i.e X(HH)=2, X(HT)=1, X(TH)=1,X(TT)=0. Thus the domain of  X is S and range is {0,1,2}.

✦ For a mathematical and rigorous definition of the random variable , let us consider the triplet (S,B,P), where S is the sample space B is the 𝞼-field of subsets in S and P is the probability function on B.

   So mathematically , a random variable is a function X(𝝎) with domain S and range (-∞ , ∞) ,such that for every real number 'a' , the event [ 𝝎 :  X(𝝎) ≤ a ] ∈ B.

✦ Random variable X can be written as P( X ≤ a ) to make probability statements about 'a'.

     For simple example given above , we should write P( X ≤ 1 ) = P{ HH,HT,TH,TT } = 3/4.

NOTATION:

✦ One dimensional random variable will be denoted by capital letters X,Y,Z,... etc. A typical outcome of the experiment ( i.e a typical elements of the sample space ) will be denoted by 𝝎 or е. Thus X(𝝎) represents the real number which the random variable X associates with the outcome 𝝎 .

✦ The value of the random variable will be denoted by small letters x,y,z,... etc. 

SOME PROPERTIES OF RANDOM VARIABLE :

✦ If  X₁ , X₂  are random variables and C is a constant , then CX₁ , X₁ + X₂ , X₁ - X₂ , X₁ * X₂  also random variable.

✦ If X is a random variable , then

• 1/X , where (1/X ) ( 𝝎 ) = ∞ if X(𝝎) = 0.
• X + 𝝎 = max[ 0 , X(𝝎) ]
• X - 𝝎 = min [ 0 , X(𝝎) ]
• | X |

are also random variables.

✦ If  X₁ , X₂  are random variables ,then (i) max[ X₁ , X₂ ] & (ii) min[ X₁ , X₂ ] are also random variables .

✦ If X is a random variable and ∱(.) is a continuous function , then ∱(X) is a random variable.

✦ If X is a random variable and ∱(.) is a increasing function , then ∱(X) is a random variable.

✦ If ∱ is a function of bounded variations on every finite interval [a,b] and X is a random variable, then ∱(X) is a random variable.

REGULA - FALSI METHOD | ENGINEERING MATHEMATICS | FINDIND ROOTS OF AN EQUATION

INTRODUCTION:

Regula-Falsi method is also used for finding approximate roots of an equation .It is very similar to bisection method.

                If you don't read about bisection method OR forgot the bisection method just click here to read now.

This is also called False Position Method.

THEORY:

As before (in Bisection Method), for a given continuous function f(x) we assume that f (a) and f (b) have opposite signs ( a = lower guess, b = upper guess). This condition satisfies the Balzano's Theorem for continuous function.

Note::Theorem (Bolzano) : If the function f(x) is continuous in [a, b] and f(a)f(b) < 0 (i.e. f(x) has opposite signs signs at a and b) then a value c ∈ (a, b) exists such that f(c) = 0.

Now after this bisection method used the midpoint of the interval [a, b] as the next iterate to converge towards the root of f(x).

A better approximation is obtained if we find the point (c, 0) where the secant line L joining the points (a, f (a)) and (b, f (b)) crosses the x-axis (see the image below). To find the value c, we write down two versions of the

slope m of the line L:

We first use points (a, f (a)) and (b, f (b)) to get equation 1 (below), and then use the points (c, 0) and (b, f (b)) to get equation 2 (below). Equating these two equation we get equation 3 (below) which is easily solved for c to get equation 4 (below):

The three possibilities are the same as before in the bisection method:

• If f (a) and f (c) have opposite signs, a zero lies in [a, c].
• If f (c) and f (b) have opposite signs, a zero lies in [c, b].
• If f (c) = 0, then the zero is c.

A short GIF for how this method works:

Working Rules:

1. Find Points x0 and x1such that x0< x1and f(x0).f( x1) < 0
2. Take the interval [x0, x1] and find next value x2= x0-f( x0).(( x1- x0)/f( x1)-f( x0))
3. If f(x2)=0 then x2is an exact root.
4. If f(x1).f(x2) < 0 then x1= x2
5. If f(x0).f(x2) <0 then x0= x2
6. Repeat the step 2-5 until f(xi)=0 or f(x0)≤ Accuracy

Solve: Find a root of an equation f(x)=x3-x-1 using False Position method

Solution:
Here x3-x-1=0

Let f(x)=x3-x-1

x	0	1	2
f(x)	-1	-1	5

1st iteration :

Here f(1)=-1<0 and f(2)=5>0

∴ Now, Root lies between x0=1 and x1=2

x2=1-(-1)⋅(2-1)/(5-(-1))

x2=1.16667

f(x2)=f(1.16667)=-0.5787<0

2nd iteration :[ FROM HERE THIS LEAVES FOR READERS AND HOPE ALL ITERATION WILL BE SOLVED ]

LASTLY WE GOT

[HERE n= No. of Iteration , x0 = value of 1st Guess point and f( x0 )= value of Fuction at x0 ]
Approximate root of the equation x3-x-1=0 using False Position method is 1.32464

Group Theory Part-1 With QNA(Theoritical Approach)

Algebraic Structure:

A non-empty set equipped one or more binary operation.   

  Ex: ( G,*) , ( S,*) etc

NOTE: Here G and S are non-empty sets and  ' * ' is called binary operations.

Group:

A group is an algebraic structure ' G ' equipped with one binary operation ' * '

i.e; (G,*) along with the following four properties.

1. Closure Property:

For every a,b ∈ G ⇒ a *b ∈ G

2.Associative Property:

For every a,b,c ∈ G ⇒ a*(b*c) =(a*b)*c

3.Identity Property:

There is an element e∈G ,such that a*e=a=e*a

Here ' e ' is the identity element in G.

4.Inverse Property:

For every a∈G , There exist an element  a^-1 ∈ G such that a*a^-1 =e =a^-1*a

Here a^-1 is known as the inverse element of a.

Abelian Groups:

A group is called an abelian group if it satisfies the commutative property.

i.e; For every a,b∈ G ⇒ a*b=b*a

Other Algebraic Structures:

Groupoid:

If any algebraic structure which equipped only one binary structure satisfies the closure property only. This is called as Groupoid.

Semi Group:

If any algebraic structure which equipped only one binary structure satisfies two property (closure and associative) only is called Semi Group.

Monoid:

If any algebraic structure which equipped only one binary structure satisfies Closure,associative and identity laws is called a monoid.

Monoids are semigroups with identity.

SubGroup:

It is a subset of a group which is itself a group and satisfy all the properties of a group.

It should form the group under the same operation of its superset.

Cyclic Group:

A group G is said to be a cyclic group if there exist an element a ∈ G  such that every element of G can be expressed as some power of a.

          If G is a group generated by 'a' we say that 'a ' is a generator of G.

     G=(a)

Note:
A cyclic group is always an abelian group but every abelian group is not a cyclic group. For instance, the rational numbers under addition is an abelian group but is not a cyclic one.

I am Posting some Easy Question below. Try to solve.(Good Luck)

IF any Doubt arise just Comment us.

1. A non empty set A is termed as an algebraic structure ________
a) with respect to binary operation *
b) with respect to ternary operation ?
c) with respect to binary operation +
d) with respect to unary operation –

2. An algebraic structure _________ is called a semigroup.
a) (P, *)
b) (Q, +, *)
c) (P, +)
d) (+, *)

3. Condition for monoid is __________
a) (a+e)=a
b) (a*e)=(a+e)
c) a=(a*(a+e)
d) (a*e)=(e*a)=a

4. A cyclic group can be generated by a/an ________ element.
a) singular
b) non-singular
c) inverse
d) multiplicative

5. How many properties can be held by a group?
a) 2
b) 3
c) 5
d) 4

6.A group (M,*) is said to be abelian if ___________
a) (x+y)=(y+x)
b) (x*y)=(y*x)
c) (x+y)=x
d) (y*x)=(x+y)

7. A monoid is called a group if _______
a) (a*a)=a=(a+c)
b) (a*c)=(a+c)
c) (a+c)=a
d) (a*c)=(c*a)=e

8.A cyclic group is always _________
a) abelian group
b) monoid
c) semigroup
d) subgroup

Answers:

1.a,2.a,3.d,4.a,5.c,6.b,7.d,8.a

INTRODUCTION TO FUNCTION | RELATION & FUNCTION

Function:

A Function is an ordered triple <∱,A,B> such that

1. A and B are sets and ∱ ⊆ A × B
2. For every x ∈A  there is some y ∈ B such that <x,y>∈∱
3.If <x,y>∈∱ and <x,z>∈∱ ,then y=z ;in other words the assignment is unique in the sense that an x ∈A is assigned  atmost one element of B.

• A is called Domain of ∱ and B is called Co-domain.

• Then we will usually write y=∱(x) ,and called " y is the image of x under ∱ "

• The set {y ∈ B :there is an x ∈A such that y=∱(x)}
  is called range of ∱ .

Types Of Function:

One to One Function:

A function f: A → B is One to One if for each element of A there is a distinct element of B. It is also known as Injective. Consider if a₁ ∈ A and b₁ ∈ B, f is defined as f:A→ B such that f (a₁) = f (b₁)

Many to One Function:

It is a function which maps two or more elements of A to the same element of set B. Two or more elements of A have the same image in B.

Onto Function:

If there exists a function for which every element of set B there is (are) preimage(s) in set A, it is Onto Function. Onto is also referred as Surjective Function.

One-One and Onto Function:

A function, f is One–One and Onto or Bijective if the function f is both One to One and Onto function. In other words, the function f associates each element of A with a distinct element of B and every element of B has a pre-image in A.

Special Types Of Function:

Identity Function:

If A = B and f (x) = x for all x ∈ A, the f is called the identity function
on A .

Constant Function:

If f (x) = c for all x ∈ A, then f is called a constant function.

Polynomial Function:

A polynomial function is defined by y =a₀ + a₁x + a₂x² + … + a_nxⁿ, where n is a non-negative integer and a₀, a₁, a₂,…, _n ∈ R.

• Constant Function: If the degree is zero, the polynomial function is a constant function.
• Linear Function: The polynomial function with degree one. Such as y = x + 1
• Quadratic Function: If the degree of the polynomial function is two, then it is a quadratic function
• Cubic Function: A cubic polynomial function is a polynomial of degree three.

Rational Function:

A rational function is any function which can be represented by a rational fraction say, f(x)/g(x) in which numerator, f(x) and denominator, g(x) are polynomial functions of x, where g(x) ≠ 0.

Modulus Function:

The absolute value of any number, c is represented in the form of |c|. If any function f: R→ R is defined by f(x) = |x|, it is known as Modulus Function. For each non-negative value of x, f(x) = x and for each negative value of x, f(x) = -x, i.e.,

f(x) = {x, if x ≥ 0; – x, if x < 0.

Signum Function:

Greatest Integer Function:

If a function f: R→ R is defined by f(x) = [x], x ∈ X. It round-off to the real number to the integer less than the number.It is also called as Floor Function.

For example: [5.12] = 5

Least Integer Function:

If a function f: R→ R is defined by f(x) = ⌈x⌉, x ∈ X. It round-off to the real number to the integer greater than the number. It is also called as Ceiling Function.

For example: [5.12] = 6

Introduction to Relation

Relation:

A relation R from A to B is a subset of the Cartesian product A x B and is derived by describing a relationship between the first element (say x) and the other element (say y) of the ordered pairs in A & B.

Representation of Relation:

A relation is represented either by Roster method or by Set-builder method. Consider an example of two sets A = {9, 16, 25} and B = {5, 4, 3, -3, -4, -5}. The relation is that the elements of A are the square of the elements of B.

1. In set-builder form, R = {(x, y): x is the square of y, x ∈ A and y ∈ B}.
2. In roster form, R = {(9, 3), (9, -3), (16, 4), (16, -4), (25, 5), (25, -5)}.

Terminologies (Terms used):

1. Image:
   for any ordered pairs, in any Cartesian product (say A × B), the second element is called the image of the first element.
2. Domain:The set of all first elements of the ordered pairs in a relation R from a set A to a set B.
3. Range:The set of all second elements in a relation R from a set A to a set B is  called Range.
4. Co-domain:
   The whole set B is called Co-domain. Range ⊆ Co-domain.

Total Number of Relations:

For two non-empty set, A and B. If the number of elements in A is h i.e., |A| = h & that of B is k i.e., |B| = k, then the number of ordered pair in the Cartesian product will be |A × B| = hk. The total number of relations is 2^hk.

Types of Relations:

There are many types of relations.

Empty Relation:

If no element of set X is related or mapped to any element of Y, then the relation R in A is an empty relation, i.e, R = Φ.

Universal Relation:

A relation R, called universal relation if each element of A is related to every element of B.

Suppose A is a set of all natural numbers and B is a set of all whole numbers. The relation between A and B is universal as every element of A is in set B.

                    Empty relation and Universal relation are sometimes called trivial relation.

Inverse Relation:

Let R be a relation from set A to set B i.e., R ∈ A × B. The relation R^-1 is said to be an Inverse relation if R^-1 from set B to A is denoted by R^-1 = {(b, a): (a, b) ∈ R}.

Identity Relation:

In Identity relation, every element of set A is related to itself only. I = {(a, a), ∈ A}.

For example, If we throw two dice, we get 36 possible outcomes, (1, 1), (1, 2), … , (6, 6). If we define a relation as R: {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}, it is an identity relation.

Reflexive Relation:

If every element of set A maps to itself, the relation is Reflexive Relation. For every a ∈ A, (a, a) ∈ R.

Symmetric Relation:

A relation R on a set A is said to be symmetric if (a, b) ∈ R then (b, a) ∈ R, for all a & b ∈ A.

Transitive Relation:

A relation in a set A is transitive if, (a, b) ∈ R, (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A

Equivalence Relation:

A relation is said to be equivalence if and only if it is Reflexive, Symmetric, and Transitive.

Ordered Relation:

Partial Ordered Relation:
A relation R⊂X ✖️ X is called partial ordering if it is Reflexive,Anti-Symmetric, and Transitive, that is
   
             (x , x) ∈ R for all x ∈ X
             (x , y) ∈ R ,(y , x) ∈ R ⇒ x = y
             (x , y) ∈ R, (y , z) ∈ R ⇒ (x , z) ∈ R

Total Ordered Relation:
When all the elements of a partial order relation are comparable, the relation is called a total order Relation.
i.e;  a,b ∈ A and  a≤ b and b≤ a
[ ≤ is called relation ]

Linear Ordered:

That is, every element is related with every element one way or the other.
A total order is also called a linear order.

PreOrdered/Quasi Ordered:

A binary relation R on a set A is a quasi order if and only if it is
(1) irreflexive, { (x , x) ∈ R for all x ∉ X }
(2) transitive.  (x , y) ∈ R, (y , z) ∈ R ⇒ (x , z) ∈ R.


(minimal/maximal element):

Let < A, ≤ > be a poset, where ≤ represents an arbitrary partial order. Then an element   b ∈ A is a minimal element of A if there is no element  a ∈ A that satisfies a ≤ b. Similarly an element b ∈ A is a maximal element of A if there is no element a ∈ A that satisfies b ≤ a.
 
(least/greatest element):

Let < A, ≤ > be a poset. Then an element b ∈ A is the least element of A if for every element a ∈ A, b ≤ a.

(well order): A total order R on a set A is a well order if every non-empty subset of A has the least element.