This document contains a lesson on sets from a mathematics class taught by Anna Korfiati. It introduces definitions of sets, elements of sets, set notation including union, intersection and difference. It also covers subsets, empty sets, Cartesian products of sets, and the natural numbers. The lesson contains examples and exercises on applying set concepts.
The document discusses sets and mathematical induction. It provides examples of using mathematical induction to prove formulas for sums of sequences, such as proving that the sum of the first n natural numbers equals n(n+1)/2, and sums of squares and cubes. It also gives exercises for students to prove similar formulas using mathematical induction.
Πρόκειται για ένα πακέτο σημειώσεων από μαθήματα συνδυαστικής, που οδηγεί σταδιακά στην βαθύτερη κατανόηση και απαλλάσσει τον φοιτητή από την ανάγκη για "λέξεις κλειδιά" και "συνταγές".
Για μια πλήρη προσέγγιση στο ζήτημα της Συνδυαστικής, μπορείτε να με βρείτε στα 6974129730 και korfiati.an@gmail.com
1) The functions g, h and their composition (goh) are defined. It is shown that goh has the form f, where f is a given function.
2) The limits needed to evaluate an expression involving f are calculated.
3) Additional limits are calculated to solve an inequality involving the limits of f.
The document contains examples of quadratic equations in standard, vertex, and factored forms. Example 1 shows the equations for a parabola opening upward with standard form y = x2 + 6x, vertex form y = x2, and factored form y = x(x). Example 2 shows the equations for a parabola opening downward with standard form y = -x2 - 2x, vertex form y = -x2, and factored form y = -x(x). The document provides examples of completing quadratic equations based on the graph of a parabola.
The document presents a series of math problems focused on counting, number facts up to 5, halving objects, and quick addition and subtraction. It includes identifying numbers up to 800, making number sentences to equal 5, halving objects, and addition and subtraction facts.
1) The document provides instructions for solving various mathematical equations and expressions involving indices.
2) Examples include solving equations for specific values of x, expressing one term in terms of others, and solving simultaneous equations.
3) Key steps shown include isolating variables, applying inverse operations, and setting expressions equal to find solutions.
The document discusses sets and mathematical induction. It provides examples of using mathematical induction to prove formulas for sums of sequences, such as proving that the sum of the first n natural numbers equals n(n+1)/2, and sums of squares and cubes. It also gives exercises for students to prove similar formulas using mathematical induction.
Πρόκειται για ένα πακέτο σημειώσεων από μαθήματα συνδυαστικής, που οδηγεί σταδιακά στην βαθύτερη κατανόηση και απαλλάσσει τον φοιτητή από την ανάγκη για "λέξεις κλειδιά" και "συνταγές".
Για μια πλήρη προσέγγιση στο ζήτημα της Συνδυαστικής, μπορείτε να με βρείτε στα 6974129730 και korfiati.an@gmail.com
1) The functions g, h and their composition (goh) are defined. It is shown that goh has the form f, where f is a given function.
2) The limits needed to evaluate an expression involving f are calculated.
3) Additional limits are calculated to solve an inequality involving the limits of f.
The document contains examples of quadratic equations in standard, vertex, and factored forms. Example 1 shows the equations for a parabola opening upward with standard form y = x2 + 6x, vertex form y = x2, and factored form y = x(x). Example 2 shows the equations for a parabola opening downward with standard form y = -x2 - 2x, vertex form y = -x2, and factored form y = -x(x). The document provides examples of completing quadratic equations based on the graph of a parabola.
The document presents a series of math problems focused on counting, number facts up to 5, halving objects, and quick addition and subtraction. It includes identifying numbers up to 800, making number sentences to equal 5, halving objects, and addition and subtraction facts.
1) The document provides instructions for solving various mathematical equations and expressions involving indices.
2) Examples include solving equations for specific values of x, expressing one term in terms of others, and solving simultaneous equations.
3) Key steps shown include isolating variables, applying inverse operations, and setting expressions equal to find solutions.
A mathematical puzzle is related to mathematical facts and objects, or whose solution needs serious mathematical arguments or calculations. A mathematical puzzle is related to mathematical facts and objects, or whose solution needs serious mathematical arguments or calculations.
This document contains instructions and examples for working with quadratic equations. It includes directions to find the roots of quadratic equations using any method, as well as assignments involving determining values of a, b, and c in quadratic equations; finding the sum and product of roots; and analyzing whether a quadratic equation can be determined from its roots or the sum and product of roots.
This document explains how to find the missing coordinate of a point on a line when the slope and one point are known. It provides examples of using the slope formula to solve for the missing x or y value. Given two points on a line and the slope, the slope formula can be used to set up an equation that can then be solved algebraically for the unknown coordinate. The document encourages practicing these types of problems and provides additional examples for the reader to work through on their own.
This document provides instructions for graphing linear equations in slope-intercept form using tables of values, discusses the components of the slope-intercept form equation (y=mx+b), gives examples of writing equations in slope-intercept form with given slopes and y-intercepts, discusses constant and zero-slope functions, and provides an example equation to graph along with assigned homework problems.
For more instructional resources, CLICK me here and DON'T FORGET TO SUBSCRIBE!
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Reference:
Nivera, G. C. (2013), Grade 7 Mathematics: Pattern and Practicalities. Don Bosco Press Inc. Makati City, Philippines.
Worried about due completion of math homework? You can count on us. In Homework1.com we have excellent infrastructure to serve you even at the most critical hours of assignment submission! Try our service today and get excellent score in math exam.
This document contains 6 addition problems with 2 possible answers each, asking the reader to point to the correct answer. The problems involve adding single-digit numbers together with the sums ranging from 3 to 10.
This document discusses solving multi-step equations. It provides examples of solving equations with more than one operation by working backwards using the order of operations. Students are instructed to undo all operations to isolate the variable. The document also discusses writing and solving equations for consecutive integers and consecutive even/odd integers. It provides an example of writing and solving an equation to find three consecutive integers with a sum of 21.
This document discusses finding the sum and product of the roots of quadratic equations. It provides the formulas for calculating the sum and product of roots without explicitly solving for the individual roots. The sum of the roots is equal to -b/a, and the product of the roots is equal to c/a, where a, b, and c are the coefficients of the quadratic equation in the standard form ax^2 + bx + c = 0. Several examples are worked out applying these formulas. The document also contains exercises asking the reader to find the sum and product of roots for additional quadratic equations using the given coefficients a, b, and c.
Юрий Гаврилин (Provectus): Time Series. Models and HacksProvectus
Спикер рассказывает про time series: что это такое, какие есть модели и алгоритмы для предсказания. Как работают на хакатонах и кэггле с time series, показывает кейс подготовки к хакатон "Ак барса»".
This document provides definitions for common math vocabulary terms including addition, subtraction, multiplication, division, comparisons using greater than, less than and equal signs, even and odd numbers, fractions, mixed numbers, digits, and the number line. It defines key terms such as sum, difference, product, and quotient using numeric examples.
This document contains notes from a math lesson that discusses linear equations. It includes examples of linear equations with points on a graph, definitions of the coefficient of x and constant term, exercises to identify these values from equations and graphs, and a final example equation.
You will learn how to get the value of a, b and c given a quadratic equations.
For more instructional resources, CLICK me here! 👇👇👇
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here! 👍👍👍
https://tinyurl.com/ycjp8r7u
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This document provides instructions for an activity where students are asked to spell words with scrabble tiles, write addition equations for the numbers on the tiles, calculate the value of each word, add the values together, and find the total value for their list of words. The document includes 5 lists with 4 words each and spaces for the students to show their work.
This document provides examples for solving quadratic equations by factoring. It explains how to solve equations of the form ax^2 + bx = 0 and ax^2 + bx + c = 0 by factoring and setting each factor equal to zero. Some example problems are worked out step-by-step, including solving 11x^2 - 13x = 8x - 3x^2 and 7x^2 + 18x = 10x^2 + 12x. The document also discusses using the fact that the roots of ax^2 + bx = 0 are x = 0 and x = -b/a to solve equations without factoring. It concludes by explaining how to use the zero product property to solve a quadratic
The document provides instructions for a Scrabble addition worksheet activity. Students are asked to arrange Scrabble tiles to spell out words, write addition equations using the numbers on the tiles, solve the equations to find the value of each word, add the values together, and find the total value for their list of words. The document provides 5 sample word lists with spaces for the student to show their work.
This document contains 7 multiple choice questions about solving absolute value equations and inequalities. The questions cover solving equations of the form |ax + b| = c and |ax + b| ≤ c for values of x. The correct answers are provided as options a-e for each question.
Least Square Plane and Leastsquare Quadric Surface Approximation by Using Mod...IOSRJM
Now a days Surface fitting is applied all engineering and medical fields. Kamron Saniee ,2007 find a simple expression for multivariate LaGrange’s Interpolation. We derive a least square plane and least square quadric surface Approximation from a given N+1 tabular points when the function is unique. We used least square method technique. We can apply this method in surface fitting also.
The document contains 15 multiple choice questions about solving systems of linear equations and inequalities. The questions ask the reader to identify the solution set for equations like |1 - 2x| >= |x - 2| and systems of equations like {x + y + z = 4, 2x + 2y - z = 5, x - y = 1}.
A mathematical puzzle is related to mathematical facts and objects, or whose solution needs serious mathematical arguments or calculations. A mathematical puzzle is related to mathematical facts and objects, or whose solution needs serious mathematical arguments or calculations.
This document contains instructions and examples for working with quadratic equations. It includes directions to find the roots of quadratic equations using any method, as well as assignments involving determining values of a, b, and c in quadratic equations; finding the sum and product of roots; and analyzing whether a quadratic equation can be determined from its roots or the sum and product of roots.
This document explains how to find the missing coordinate of a point on a line when the slope and one point are known. It provides examples of using the slope formula to solve for the missing x or y value. Given two points on a line and the slope, the slope formula can be used to set up an equation that can then be solved algebraically for the unknown coordinate. The document encourages practicing these types of problems and provides additional examples for the reader to work through on their own.
This document provides instructions for graphing linear equations in slope-intercept form using tables of values, discusses the components of the slope-intercept form equation (y=mx+b), gives examples of writing equations in slope-intercept form with given slopes and y-intercepts, discusses constant and zero-slope functions, and provides an example equation to graph along with assigned homework problems.
For more instructional resources, CLICK me here and DON'T FORGET TO SUBSCRIBE!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
Reference:
Nivera, G. C. (2013), Grade 7 Mathematics: Pattern and Practicalities. Don Bosco Press Inc. Makati City, Philippines.
Worried about due completion of math homework? You can count on us. In Homework1.com we have excellent infrastructure to serve you even at the most critical hours of assignment submission! Try our service today and get excellent score in math exam.
This document contains 6 addition problems with 2 possible answers each, asking the reader to point to the correct answer. The problems involve adding single-digit numbers together with the sums ranging from 3 to 10.
This document discusses solving multi-step equations. It provides examples of solving equations with more than one operation by working backwards using the order of operations. Students are instructed to undo all operations to isolate the variable. The document also discusses writing and solving equations for consecutive integers and consecutive even/odd integers. It provides an example of writing and solving an equation to find three consecutive integers with a sum of 21.
This document discusses finding the sum and product of the roots of quadratic equations. It provides the formulas for calculating the sum and product of roots without explicitly solving for the individual roots. The sum of the roots is equal to -b/a, and the product of the roots is equal to c/a, where a, b, and c are the coefficients of the quadratic equation in the standard form ax^2 + bx + c = 0. Several examples are worked out applying these formulas. The document also contains exercises asking the reader to find the sum and product of roots for additional quadratic equations using the given coefficients a, b, and c.
Юрий Гаврилин (Provectus): Time Series. Models and HacksProvectus
Спикер рассказывает про time series: что это такое, какие есть модели и алгоритмы для предсказания. Как работают на хакатонах и кэггле с time series, показывает кейс подготовки к хакатон "Ак барса»".
This document provides definitions for common math vocabulary terms including addition, subtraction, multiplication, division, comparisons using greater than, less than and equal signs, even and odd numbers, fractions, mixed numbers, digits, and the number line. It defines key terms such as sum, difference, product, and quotient using numeric examples.
This document contains notes from a math lesson that discusses linear equations. It includes examples of linear equations with points on a graph, definitions of the coefficient of x and constant term, exercises to identify these values from equations and graphs, and a final example equation.
You will learn how to get the value of a, b and c given a quadratic equations.
For more instructional resources, CLICK me here! 👇👇👇
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here! 👍👍👍
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
This document provides instructions for an activity where students are asked to spell words with scrabble tiles, write addition equations for the numbers on the tiles, calculate the value of each word, add the values together, and find the total value for their list of words. The document includes 5 lists with 4 words each and spaces for the students to show their work.
This document provides examples for solving quadratic equations by factoring. It explains how to solve equations of the form ax^2 + bx = 0 and ax^2 + bx + c = 0 by factoring and setting each factor equal to zero. Some example problems are worked out step-by-step, including solving 11x^2 - 13x = 8x - 3x^2 and 7x^2 + 18x = 10x^2 + 12x. The document also discusses using the fact that the roots of ax^2 + bx = 0 are x = 0 and x = -b/a to solve equations without factoring. It concludes by explaining how to use the zero product property to solve a quadratic
The document provides instructions for a Scrabble addition worksheet activity. Students are asked to arrange Scrabble tiles to spell out words, write addition equations using the numbers on the tiles, solve the equations to find the value of each word, add the values together, and find the total value for their list of words. The document provides 5 sample word lists with spaces for the student to show their work.
This document contains 7 multiple choice questions about solving absolute value equations and inequalities. The questions cover solving equations of the form |ax + b| = c and |ax + b| ≤ c for values of x. The correct answers are provided as options a-e for each question.
Least Square Plane and Leastsquare Quadric Surface Approximation by Using Mod...IOSRJM
Now a days Surface fitting is applied all engineering and medical fields. Kamron Saniee ,2007 find a simple expression for multivariate LaGrange’s Interpolation. We derive a least square plane and least square quadric surface Approximation from a given N+1 tabular points when the function is unique. We used least square method technique. We can apply this method in surface fitting also.
The document contains 15 multiple choice questions about solving systems of linear equations and inequalities. The questions ask the reader to identify the solution set for equations like |1 - 2x| >= |x - 2| and systems of equations like {x + y + z = 4, 2x + 2y - z = 5, x - y = 1}.
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
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Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
Gender and Mental Health - Counselling and Family Therapy Applications and In...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
2. 1. Σύνολα
Άννα Κορφιάτη | korfiati.an@gmail.com | https://www.facebook.com/mathwithcoding | 6974 12 9730
Ορισμός (Cantor). Σύνολο είναι κάθε συλλογή αντικειμένων που γίνονται
αντιληπτά διά της εμπειρίας μας ή της διανόησής μας, και είναι σαφώς
καθορισμένα και διακεκριμένα μεταξύ τους.
Τα επί μέρους αντικείμενα της συλλογής ονομάζονται στοιχεία ή μέλη
του συνόλου.
Ορισμός. Ονομάζουμε κενό σύνολο ∅ ένα σύνολο χωρίς στοιχεία
3. 1. Σύνολα
Άννα Κορφιάτη | korfiati.an@gmail.com | https://www.facebook.com/mathwithcoding | 6974 12 9730
Σύνολα: κεφαλαία γράμματα Χ, Υ, Ζ
Στοιχεία συνόλου: μικρά γράμματα x,y,z,a,b
Tο σύνολο X αποτελείται από τα στοιχεία 𝒙, 𝒚, 𝒛, …
𝑋 = 𝑥, 𝑦, 𝑧, …
Ανήκειν: το αντικείμενο 𝑥 (δεν) είναι στοιχείο του συνόλου X
𝑥 ∈ 𝑋 (𝑥 ∉ 𝑋)
Ποσοδείκτες:
Καθολικός ποσοδείκτης: Για κάθε «∀»
Υπαρκτικός ποσοδείκης: Υπάρχει τουλάχιστον ένα «∃»
Συνεπαγωγή και ισοδυναμία
Υποθετική πρόταση: Αν … τότε ... «⇒»
… αν και μόνο αν (τότε και μόνο τότε) «⇔»
Όταν μία ισότητα, ή μία ισοδυναμία τίθεται εξ ορισμού, γράφουμε
αντίστοιχα
«≔» «: ⇔»
4. 1. Σύνολα
Άννα Κορφιάτη | korfiati.an@gmail.com | https://www.facebook.com/mathwithcoding | 6974 12 9730
Αν ο Μάνος είναι άνθρωπος τότε ο Μάνος είναι θνητός
Ο Μάνος είναι άνθρωπος αν και μόνον αν ο Μάνος είναι θνητός
6. 1. Σύνολα
Άννα Κορφιάτη | korfiati.an@gmail.com | https://www.facebook.com/mathwithcoding | 6974 12 9730
Με αναγραφή των στοιχείων του συνόλου
Α = 1,2,3
Με αναγραφή της συνθήκης ή της χαρακτηριστικής
ιδιότητας των στοιχείων του συνόλου
Α = 𝑥: 𝑥 𝛼𝜌𝜏𝜄𝜊𝜍 απο το 1 𝜔𝜍 𝜏𝜊 1000
Γενικότερα αν 𝑃 𝑥 είναι η χαρακτηριστική ιδιότητα που
ικανοποιεί κάθε στοιχείο 𝑥 ενός συνόλου Α , θα
γράφουμε,
Α = 𝑥: 𝑃 𝑥
Καθορισμός συνόλου
13. 1. Σύνολα
Άννα Κορφιάτη | korfiati.an@gmail.com | https://www.facebook.com/mathwithcoding | 6974 12 9730
• Ορισµός 4. Έστω δύο σύνολα 𝐴 και 𝐵.
1. Τοµή των 𝐴 και 𝐵 καλείται το σύνολο που αποτελείται από τα κοινά στοιχεία των δύο
συνόλων και συµβολίζεται ως 𝛢 ∩ 𝐵.
2. Ενωση των 𝐴 και 𝐵 καλείται το σύνολο που αποτελείται από τα στοιχεία και των δυο
συνόλων και συµβολίζεται ως 𝐴 ∪ 𝐵.
3. ∆ιαφορά του 𝐵 από το 𝐴, µε αυτή τη σειρά, καλείται το σύνολο που αποτελείται από
εκείνα τα στοιχεία του 𝐵 τα οποία δεν ανήκουν στο 𝐴 και συµβολίζεται ως 𝐵A ή 𝐵 − A
4. Αν το σύνολο 𝐴 είναι υποσύνολο ενός σύνολου Χ ορίζουμε το συμπλήρωμα 𝑨𝒄
του
συνόλου 𝐴,
𝐴𝑐
≔ 𝑥 ∈ 𝑋: 𝑥 ∉ 𝐴 = 𝑋A
22. 1. Σύνολα
Άννα Κορφιάτη | korfiati.an@gmail.com | https://www.facebook.com/mathwithcoding | 6974 12 9730
• Για δύο μη κενά σύνολα 𝐴, 𝐵 ορίζουμε το καρτεσιανό γινόμενο 𝐴 × 𝐵
ως το σύνολο των διατεταγμένων ζευγών (𝛼, 𝛽) με πρώτο στοιχείο 𝛼
από το σύνολο 𝐴 και δεύτερο στοιχείο 𝛽 από το 𝐵
𝐴 × 𝐵 ≔ { 𝛼, 𝛽 : 𝛼 ∈ 𝐴 και 𝛽 ∈ Β}
23. 1. Σύνολα
Άννα Κορφιάτη | korfiati.an@gmail.com | https://www.facebook.com/mathwithcoding | 6974 12 9730
Οι φυσικοί αριθμοί
• Διαισθητικά: Οι απλούστεροι αριθμοί είναι οι «αριθμοί απαριθμησης»
1,2,3, …
Οι αριθμοί αυτοί ονομάζονται φυσικοί αριθμοί.
Το σύνολο των φυσικών αριθμών συμβολίζεται με ℕ∗
.
ℕ∗
≔ 1,2,3, …
24. 1. Σύνολα
Άννα Κορφιάτη | korfiati.an@gmail.com | https://www.facebook.com/mathwithcoding | 6974 12 9730
• Πρόσθεση: απεικόνιση ℕ∗
× ℕ∗
→ ℕ∗
όπου σε κάθε ζεύγος φυσικών αριθμών (𝜈, 𝜇)
αντιστοιχεί ακριβώς ένας φυσικός αριθμός 𝜅:
𝜈, 𝜇 ↦ 𝜅 = 𝜈 + 𝜇
• Η πρόσθεση στους φυσικούς αριθμούς έχει τις ακόλουθες ιδιότητες:
1. Αν α = 𝛽, τότε α + γ = 𝛽 + 𝛾
2. α + β = β + α
3. α + (β + γ) = (𝛼 + 𝛽) + 𝛾
• Η εξίσωση 𝛼 + 𝑥 = 𝛽 δεν έχει λύση για κάθε ζεύγος φυσικών αριθμών α, 𝛽.
26. 1. Σύνολα
Άννα Κορφιάτη | korfiati.an@gmail.com | https://www.facebook.com/mathwithcoding | 6974 12 9730
• Πολλαπλασιασμός: απεικόνιση ℕ∗
× ℕ∗
→ ℕ∗
όπου σε κάθε ζεύγος φυσικών
αριθμών (𝜈, 𝜇) αντιστοιχεί ακριβώς ένας φυσικός αριθμός 𝜅:
𝜈, 𝜇 ↦ 𝜅 = 𝜈𝜇
• Η πρόσθεση στους φυσικούς αριθμούς έχει τις ακόλουθες ιδιότητες:
1. Αν α = 𝛽, τότε αγ = 𝛽𝛾
2. αβ = βα
3. α(βγ) = (𝛼𝛽)𝛾
• Η εξίσωση 𝛼𝑥 = 𝛽 δεν έχει λύση για κάθε ζεύγος φυσικών αριθμών α, 𝛽.
• Επιμεριστική ιδιότητα:
α + β γ = 𝛼𝛾 + 𝛽𝛾
27. 1. Σύνολα
Άννα Κορφιάτη | korfiati.an@gmail.com | https://www.facebook.com/mathwithcoding | 6974 12 9730
• Μια θεωρία αποτελείται από μια ολιγοπληθή ομάδα κατάλληλων
πρωταρχικών όρων, οι οποίοι δεν ορίζονται, καθώς και από ολιγοπληθή
ομάδα κατάλληλων προτάσεων που δίδονται χωρίς απόδειξη, αξιωμάτων.
• Κάθε άλλος όρος της θεωρίας ορίζεται με βάση τους πρωταρχικούς όρους.
• Κάθε άλλη πρόταση της θεωρίας, θεώρημα, αποδεικνύεται από τα αξιώματα
στη βάση συμφωνημένων κανόνων συλλογισμού.
• Τα αξιώματα αποτελούν έμμεσους ορισμούς των πρωταρχικών όρων.
• Ένα σύστημα αξιωμάτων πρέπει να είναι μη αντιφατικό.
• Τα αξιώματα θα πρέπει να είναι ανεξάρτητα.
• Η βασική διαφορά της σύγχρονης αξιωματικής μεθόδου από την παραδοσιακή
(Ευκλείδης) συνίσταται στο ότι είναι απαλλαγμένη από κάθε προσφυγή στην
εμπειρική εποπτεία την οποία έχουμε για τους πρωταρχικούς όρους ή για την
προφανή αλήθεια των αξιωμάτων.
Η σύγχρονη αξιωματική μέθοδος
29. 1. Σύνολα
Άννα Κορφιάτη | korfiati.an@gmail.com | https://www.facebook.com/mathwithcoding | 6974 12 9730
Τα αξιώματα του Peano
Ρ1 : Το ένα είναι αριθμός.
Ρ2 : Ο διάδοχος αριθμού είναι πάλι αριθμός
Ρ3 : Το ένα δεν είναι διάδοχος αριθμού
Ρ4 : Δεν υπάρχουν διαφορετικοί αριθμοί με τον ίδιο διάδοχο.
Ρ5 : Κάθε ιδιότητα την οποία έχει το ένα και η οποία ανήκει στον διάδοχο κάθε
αριθμού που έχει αυτή την ιδιότητα, ανήκει σε όλους τους αριθμούς.
30. 1. Σύνολα
Άννα Κορφιάτη | korfiati.an@gmail.com | https://www.facebook.com/mathwithcoding | 6974 12 9730
Τα αξιώματα του Peano
Το σύνολο ℕ∗ονομάζεται σύνολο των φυσικών αριθμών αν και μόνο αν,
Ρ1 : 1 ∈ ℕ∗
Ρ2 : ∀𝜈 ∈ ℕ∗
⇒ 𝜈 + 1 ∈ ℕ∗
Ρ3 : ∀𝜈 ∈ ℕ∗ ⇒ 𝜈 + 1 ≠ 1
Ρ4 : 𝜈 ∈ ℕ∗
και μ ∈ ℕ∗
και 𝜈 + 1 = 𝜇 + 1 ⇒ 𝜈 = 𝜇
Ρ5 : Αν A ⊆ ℕ∗
και 1 ∈ Α και ∀𝜈 ∈ Α ⇒ 𝜈 + 1 ∈ Α τότε Α = ℕ∗