Expressions Of Pj Problems

Pj Problems - Overview

Celestial Stars

The Number Line

Geometries

7 Spaces Of Interest - Overview

Triadic Unit Mesh

Creation

The Atom

Survival

Energy

Light

Heat

Sound

Music

Language

Stories

Work

States Of Matter

Buoyancy

Nuclear Reactions

Molecular Shapes

Electron Configurations

Chemical Bonds

Energy Conversion

Chemical Reactions

Electromagnetism

Continuity

Growth

Human-cells

Proteins

Nucleic Acids

COHN - Natures Engineering Of The Human Body

The Human-Body Systems

Vision

Walking

Behaviors

Sensors Sensings

Beauty

Faith, Love, Charity

Photosynthesis

Weather

Systems

Algorithms

Tools

Networks

Search

Differential Calculus

Antiderivative

Integral Calculus

Economies

Inflation

Markets

Money Supply

Painting

Uniqueness Proof Technique

Suppose *If A then B* is a proposition involving statements A and B. The existence of B does not necessarily establish the uniqueness of B. The Uniqueness Proof Technique can be used to establish the uniqueness of B.

Prove, by the direct uniqueness method, that if a, b, c, d, e, and f are real numbers such that (ad - bc) ≠ 0, then there are unique real numbers x and y such that (ax + by) = e and (cx + dy) = f

**The strings**:
S_{7}P_{2}A_{21} (Identity - Physical Property).
**The math**:

Pj Problem of Interest is of type *identity* (physical property). Proofs establish truths. So they are *identity* problems.

The conceptual steps used in the Uniqueness Proof Technique is illustrated in figure 121.4.
*Existence* is usually explicitly indicated with the *quantifiers*: *there is, there are, for all, for each, etc*. These *quantifiers* can be assigned to two main groups:*existential quantifiers*(e.g. there is, there are) and *universal quantifiers* (e.g. for all).

Establish the existence of x and y in the given problem:

The *existential quantifier, "there are" * is implicit in the problem. In order words, we can restate B as:

There are real numbers x and y such that (ax + by) = e and (cx + dy) = f.

If we multiply (ax + by) = e by d, we have:

d(ax + by) = de ------(1)

If we multiply (cx + dy) by b, we have:

b(cx + dy) = bf ------(2)

subtracting equation (2) from (1), we have:

(dax + dby - bcx -bdy) = de - bf--------(3)

So, (da - bc)x + (db - bd)y = de - bf

So, x = (de - bf)/(da - bc) ------------(4)

Similarly, if we multiply (ax + by) = e by c and (cx + dy) by a, we have:

y = (af - ce)/(ad - bc) ----------------(5)

The values of x and y exist for the given problem if they satisfy:

(ax + by) = e and (cx + dy)

[a(de-bf)/(da-bc) + b(af-ce)/(ad-bc)] = [ade -abf + baf - bce]/(da-bc)

= e(ad-bc)/(da-bc) = e.

[c(de-bf)/(da-bc) + d(af-ce)/(ad-bc)] = [cde -cbf + daf - dce]/(da-bc)

= f(ad-bc)/(da-bc) = f.

So, existence of x and y established.

Establish the uniqueness of x and y in the given problem:

Assume two different x values (x_{1}, x_{2}), and two different y values(y_{1}, y_{2}) then:

ax_{1} + by_{1} = e-------(1)

cx_{1} + dy_{1} = f-------(2)

ax_{2} + by_{2} = e-------(3)

cx_{2} + dy_{2} = f-------(4)

equation(1) - Equation(3) gives:

[a(x_{1}-x_{2}) + b(y_{1}-y_{2})] = 0 -------(5)

equation(2) - Equation(4) gives:

[c(x_{1}-x_{2}) + d(y_{1}-y_{2})] = 0---------(6)

Multiplying equation(5) by d and equation (6) by b, then subtracting (6) from (5) gives:

[(ad - bc)(x_{1}-x_{2})] = 0.

So, (x_{1}-x_{2}) = 0 since (ad - bc) ≠ 0.

So, x_{1} = x_{2}

Similarly, y_{1} = y_{2}

So, uniqueness is proved.

Math

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

Derivation Of The Area Of A Circle, A Sector Of A Circle And A Circular Ring

Derivation Of The Area Of A Trapezoid, A Rectangle And A Triangle

Derivation Of The Area Of An Ellipse

Derivation Of Volume Of A Cylinder

Derivation Of Volume Of A Sphere

Derivation Of Volume Of A Cone

Derivation Of Volume Of A Torus

Derivation Of Volume Of A Paraboloid

Volume Obtained By Revolving The Curve y = x^{2} About The X Axis

Single Variable Functions

Absolute Value Functions

Conics

Real Numbers

Vector Spaces

Equation Of The Ascent Path Of An Airplane

Calculating Capacity Of A Video Adapter Board Memory

Probability Density Functions

Boolean Algebra - Logic Functions

Ordinary Differential Equations (ODEs)

Infinite Sequences And Series

Introduction To Group Theory

Advanced Calculus - Partial Derivatives

Advanced Calculus - General Charateristics Of Partial Differential Equations

Advanced Calculus - Jacobians

Advanced Calculus - Solving PDEs By The Method Of Separation Of Variables

Advanced Calculus - Fourier Series

Advanced Calculus - Multiple Integrals

Production Schedule That Maximizes Profit Given Constraint Equation

Separation Of Variables As Solution Method For Homogeneous Heat Flow Equation

Newton And Fourier Cooling Laws Applied To Heat Flow Boundary Conditions

Fourier Series

Derivation Of Heat Equation For A One-Dimensional Heat Flow

Homogenizing-Non-Homogeneous-Time-Varying-IBVP-Boundary-Condition

The Universe is composed of *matter* and *radiant energy*. *Matter* is any kind of *mass-energy* that moves with velocities less than the velocity of light. *Radiant energy* is any kind of *mass-energy* that moves with the velocity of light.

Periodic Table

Composition And Structure Of Matter

How Matter Gets Composed

How Matter Gets Composed (2)

Molecular Structure Of Matter

Molecular Shapes: Bond Length, Bond Angle

Molecular Shapes: Valence Shell Electron Pair Repulsion

Molecular Shapes: Orbital Hybridization

Molecular Shapes: Sigma Bonds Pi Bonds

Molecular Shapes: Non ABn Molecules

Molecular Orbital Theory

More Pj Problem Strings