1.4 Abstract objects and expressions
•
1 like•209 views
Selected items from the Introduction to Set Theory and to Methodology and Philosophy of Mathematics and Computer Programming.
Report
Share
Report
Share
Download to read offline
Recommended
6.3 Equivalences versus partitions
This document introduces equivalence relations and partitions. It defines an equivalence relation as a binary relation that is reflexive, symmetric, and transitive. Equivalence relations partition a set into disjoint equivalence classes that cover the entire set. The quotient set of a set by an equivalence relation consists of the equivalence classes. Every equivalence relation determines a partition, and every partition determines an equivalence relation. Examples are provided to illustrate these concepts using the equivalence relation of congruence modulo 3 on the integers.
6.1 Partitions
Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.
6.2 Reflexivity, symmetry and transitivity (dynamic slides)
This document introduces basic concepts of set theory including definitions of reflexive, symmetric and transitive relations. It provides examples of relations between lines that are parallel or perpendicular. It also discusses relations between people like siblings and ancestors. The document proves properties of congruence relations and discusses counting relations with certain properties over sets with a given number of elements. It addresses a claim about relations that are symmetric and transitive necessarily being reflexive, finding a counterexample.
6.2 Reflexivity, symmetry and transitivity (handout)
Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.
5.8 Permutations (handout)
Selected items from the introduction to set theory and to methodology and philosophy of mathematics and computer programming.
5.8 Permutations (dynamic slides)
This document provides an introduction to permutations. It begins by defining a permutation as a bijection from a set to itself. It then gives examples of permutations of small sets. The document proves several properties of permutations, including that the number of permutations of a set with n elements is n!. It introduces the concept of powers of permutations and proves related properties. It concludes by proving additional properties of permutations using an algebraic approach that relies on basic properties like associativity of composition.
5.7 Function powers
Selected items from set theory and from methodology and philosophy of mathematics and computer programming.
5.6 Function inverse. A handout.
Selected items from set theory and methodology and philosophy of mathematics and computer programming.
Recommended
6.3 Equivalences versus partitions
This document introduces equivalence relations and partitions. It defines an equivalence relation as a binary relation that is reflexive, symmetric, and transitive. Equivalence relations partition a set into disjoint equivalence classes that cover the entire set. The quotient set of a set by an equivalence relation consists of the equivalence classes. Every equivalence relation determines a partition, and every partition determines an equivalence relation. Examples are provided to illustrate these concepts using the equivalence relation of congruence modulo 3 on the integers.
6.1 Partitions
Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.
6.2 Reflexivity, symmetry and transitivity (dynamic slides)
This document introduces basic concepts of set theory including definitions of reflexive, symmetric and transitive relations. It provides examples of relations between lines that are parallel or perpendicular. It also discusses relations between people like siblings and ancestors. The document proves properties of congruence relations and discusses counting relations with certain properties over sets with a given number of elements. It addresses a claim about relations that are symmetric and transitive necessarily being reflexive, finding a counterexample.
6.2 Reflexivity, symmetry and transitivity (handout)
Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.
5.8 Permutations (handout)
Selected items from the introduction to set theory and to methodology and philosophy of mathematics and computer programming.
5.8 Permutations (dynamic slides)
This document provides an introduction to permutations. It begins by defining a permutation as a bijection from a set to itself. It then gives examples of permutations of small sets. The document proves several properties of permutations, including that the number of permutations of a set with n elements is n!. It introduces the concept of powers of permutations and proves related properties. It concludes by proving additional properties of permutations using an algebraic approach that relies on basic properties like associativity of composition.
5.7 Function powers
Selected items from set theory and from methodology and philosophy of mathematics and computer programming.
5.6 Function inverse. A handout.
Selected items from set theory and methodology and philosophy of mathematics and computer programming.
5.6 Function inverse. Dynamic slides.
Selected items from set theory and methodology and philosophy of mathematics and computer programming.
5.5 Injective and surjective functions. A handout.
Selected items from set theory and methodology and philosophy of mathematics and computer programming.
5.5 Injective and surjective functions. Dynamic slides.
Selected items from set theory and from methodology and philosophy of mathematics and computer programming.
5.3 Basic functions. A handout.
Selected items from the set theory and methodology and philosophy of mathematics and computer programming.
5.3 Basic functions. Dynamic slides.
Selected items from the set theory and methodology and philosophy of mathematics and computer programming.
5.1 Defining and visualizing functions. A handout.
This document introduces concepts related to functions including:
- Defining functions in terms of unique mappings between inputs and outputs
- Distinguishing between total, partial, and non-functions
- Specifying domains and ranges
- Using vertical line tests to identify functions from graphs
- Examples of functions defined by formulas or mappings
5.1 Defining and visualizing functions. Dynamic slides.
Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.
1.8 Separation schema
Selected items from the Introduction to Set Theory and to Methodology and Philosophy of Mathematics and Computer Programming.
1.2 Axiom of pair
Selected items from the Introduction to Set Theory and to Methodology and Philosophy of Mathematics and Computer Programming.
1.11 Mathematical induction
Selected items from Introduction to Set Theory and Methodology and Philosophy of Mathematics and Computer Programming.
1.7 The sets of numbers
Selected items from Introduction to Set Theory and Methodology and Philosophy of Mathematics and Computer Programming.
1.6 Subsets
Selected items from Introduction to Set Theory and Methodology and Philosophy of Mathematics and Computer Programming.
1.1 Notions of set and membership
Selected items from Introduction to Set Theory and Methodology and Philosophy of Mathematics and Computer Programming.
5.2. Function composition
Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programing.
4.7 Powers of binary relations
Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming
4.6 Relative product and composition
Selected items from the Introduction to set theory and to me methodology and philosophy of mathematics and computer programing.
4.5 Inverse relation
This document introduces the concept of an inverse relation. It defines an inverse relation R-1 as the set containing all pairs (b,a) such that (a,b) is in R. It provides examples of inverse relations and shows how inverse relations can be defined using formulas, tuples, truth tables, and visual representations like graphs. The document also presents exercises for identifying inverse relations and propositions about inverse relations and their properties.
4.4 Set operations on relations
Selected items form the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.
4.1 Defining and visualizing binary relations
Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.
3.7 Indexed families of sets
Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.
More Related Content
More from Jan Plaza
5.6 Function inverse. Dynamic slides.
Selected items from set theory and methodology and philosophy of mathematics and computer programming.
5.5 Injective and surjective functions. A handout.
Selected items from set theory and methodology and philosophy of mathematics and computer programming.
5.5 Injective and surjective functions. Dynamic slides.
Selected items from set theory and from methodology and philosophy of mathematics and computer programming.
5.3 Basic functions. A handout.
Selected items from the set theory and methodology and philosophy of mathematics and computer programming.
5.3 Basic functions. Dynamic slides.
Selected items from the set theory and methodology and philosophy of mathematics and computer programming.
5.1 Defining and visualizing functions. A handout.
This document introduces concepts related to functions including:
- Defining functions in terms of unique mappings between inputs and outputs
- Distinguishing between total, partial, and non-functions
- Specifying domains and ranges
- Using vertical line tests to identify functions from graphs
- Examples of functions defined by formulas or mappings
5.1 Defining and visualizing functions. Dynamic slides.
Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.
1.8 Separation schema
Selected items from the Introduction to Set Theory and to Methodology and Philosophy of Mathematics and Computer Programming.
1.2 Axiom of pair
Selected items from the Introduction to Set Theory and to Methodology and Philosophy of Mathematics and Computer Programming.
1.11 Mathematical induction
Selected items from Introduction to Set Theory and Methodology and Philosophy of Mathematics and Computer Programming.
1.7 The sets of numbers
Selected items from Introduction to Set Theory and Methodology and Philosophy of Mathematics and Computer Programming.
1.6 Subsets
Selected items from Introduction to Set Theory and Methodology and Philosophy of Mathematics and Computer Programming.
1.1 Notions of set and membership
Selected items from Introduction to Set Theory and Methodology and Philosophy of Mathematics and Computer Programming.
5.2. Function composition
Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programing.
4.7 Powers of binary relations
Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming
4.6 Relative product and composition
Selected items from the Introduction to set theory and to me methodology and philosophy of mathematics and computer programing.
4.5 Inverse relation
This document introduces the concept of an inverse relation. It defines an inverse relation R-1 as the set containing all pairs (b,a) such that (a,b) is in R. It provides examples of inverse relations and shows how inverse relations can be defined using formulas, tuples, truth tables, and visual representations like graphs. The document also presents exercises for identifying inverse relations and propositions about inverse relations and their properties.
4.4 Set operations on relations
Selected items form the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.
4.1 Defining and visualizing binary relations
Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.
3.7 Indexed families of sets
Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.
More from Jan Plaza (20)
5.5 Injective and surjective functions. A handout.
5.5 Injective and surjective functions. A handout.
5.5 Injective and surjective functions. Dynamic slides.
5.5 Injective and surjective functions. Dynamic slides.
5.1 Defining and visualizing functions. A handout.
5.1 Defining and visualizing functions. A handout.
5.1 Defining and visualizing functions. Dynamic slides.
5.1 Defining and visualizing functions. Dynamic slides.
Recently uploaded
Azure Interview Questions and Answers PDF By ScholarHat
Azure Interview Questions and Answers PDF By ScholarHat
Digital Artifact 1 - 10VCD Environments Unit
Digital Artifact 1 - 10VCD Environments Unit - NGV Pavilion Concept Design
Community pharmacy- Social and preventive pharmacy UNIT 5
Covered community pharmacy topic of the subject Social and preventive pharmacy for Diploma and Bachelor of pharmacy
Walmart Business+ and Spark Good for Nonprofits.pdf
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
PCOS corelations and management through Ayurveda.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Aberdeen
BBR 2024 Summer Sessions Interview Training
Qualitative research interview training by Professor Katrina Pritchard and Dr Helen Williams
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
-------------------------------------------------------------------------------
Find out more about ISO training and certification services
Training: ISO/IEC 27001 Information Security Management System - EN | PECB
ISO/IEC 42001 Artificial Intelligence Management System - EN | PECB
General Data Protection Regulation (GDPR) - Training Courses - EN | PECB
Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
-------------------------------------------------------------------------------
For more information about PECB:
Website: https://pecb.com/
LinkedIn: https://www.linkedin.com/company/pecb/
Facebook: https://www.facebook.com/PECBInternational/
Slideshare: http://www.slideshare.net/PECBCERTIFICATION
Executive Directors Chat Leveraging AI for Diversity, Equity, and Inclusion
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
How to Fix the Import Error in the Odoo 17
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...National Information Standards Organization (NISO)
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.The History of Stoke Newington Street Names
Presented at the Stoke Newington Literary Festival on 9th June 2024
www.StokeNewingtonHistory.com
Recently uploaded (20)
Digital Artefact 1 - Tiny Home Environmental Design
Digital Artefact 1 - Tiny Home Environmental Design
Liberal Approach to the Study of Indian Politics.pdf
Liberal Approach to the Study of Indian Politics.pdf
Azure Interview Questions and Answers PDF By ScholarHat
Azure Interview Questions and Answers PDF By ScholarHat
Community pharmacy- Social and preventive pharmacy UNIT 5
Community pharmacy- Social and preventive pharmacy UNIT 5
Walmart Business+ and Spark Good for Nonprofits.pdf
Walmart Business+ and Spark Good for Nonprofits.pdf
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
Executive Directors Chat Leveraging AI for Diversity, Equity, and Inclusion
Executive Directors Chat Leveraging AI for Diversity, Equity, and Inclusion
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...