How to Write Proofs in Set Theory with the Math Sorcerer from Udemy

This is a course on Some simple examples are given at the beginning, but soon after, the PROOFS begin.

Important Note: A formal prerequisite for this material is an understanding of mathematical logic. However I have tried to explain the ideas from logic as they come up in the proofs.

The best way to learn to write proofs is to watch someone else give careful proofs and then try to do it on your own.

This is the MANTRA behind this course which is full of beautiful 100% correct well written proofs.

Basically just,

1) Watch the introduction videos if you need to. These cover the very basic things that you should know regarding sets.

2) Watch the PROOF videos in ANY order. Take notes, try to understand them.

3) Try to do the proofs on your own. If you can't, watch the video and cheat and try again.

It takes A LOT of effort to learn to write proofs like the ones you see in these videos, so don't feel discouraged if you find it difficult, it is supposed to be tough.

This is an I hope you enjoy watching these videos and working through these problems as much as I have:)

Note this course has lots of very short videos. If you are trying to learn math then this format can be good because you don't have to spend tons of time on the course every day. Even if you can only spend time doing 1 video a day, that is honestly better than not doing any mathematics. You can learn a lot and because there are so many videos you could do 1 video a day for a very long time. Remember that math can be challenging and time consuming, so if you just do a little bit every day it can make your journey much more enjoyable. I hope you enjoy this course and learn lots of mathematics.

What's inside

Learning objectives

How to write mathematical proofs with sets
The mathematical notation surrounding sets
The definition of a set and subset
The definition of union, intersection, relative complement(set minus), and absolute complement
How to show one set is a subset of another
How to show two sets are equal
How to use various proof techniques in set theory proofs
How to use the method of double inclusion
How to prove sets are equal without the method of double inclusion

How to do a proof by contradiction
Understand the common sets found in mathematics including the natural numbers, integers, rational numbers, real numbers, and complex numbers
Understand what empty set actually is
Understand venn diagrams
Understand the powerset and how to find it
Understand cardinality(bonus section, more advanced)
How to prove two sets have the same cardinality(bonus section, more advanced)
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How to write mathematical proofs with sets
The mathematical notation surrounding sets
The definition of a set and subset
The definition of union, intersection, relative complement(set minus), and absolute complement
How to show one set is a subset of another
How to show two sets are equal
How to use various proof techniques in set theory proofs
How to use the method of double inclusion
How to prove sets are equal without the method of double inclusion
How to do a proof by contradiction
Understand the common sets found in mathematics including the natural numbers, integers, rational numbers, real numbers, and complex numbers
Understand what empty set actually is
Understand venn diagrams
Understand the powerset and how to find it
Understand cardinality(bonus section, more advanced)
How to prove two sets have the same cardinality(bonus section, more advanced)
Show more
Show less

Syllabus

Introduction to Set Theory

Introduction

Set Operations: Union, Intersection, Set Minus, and Absolute Complement

Simple Example of Set Operations

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Read about what's good

what should give you pause

and possible dealbreakers

Teaches proof techniques like double inclusion and contradiction, which are fundamental in advanced mathematics courses

Explores set theory, which provides a foundation for understanding more advanced topics in mathematics and computer science

Requires an understanding of mathematical logic, so learners may need to acquire this knowledge before or during the course

Focuses on writing proofs, which is a skill that takes time and effort to develop, so learners should be prepared for a challenge

Covers common sets like natural, integer, rational, real, and complex numbers, which are essential for mathematical literacy

Reviews summary

Learning to write set theory proofs

According to learners, this course is a highly effective way to learn how to write mathematical proofs in set theory. Students appreciate how the instructor, the Math Sorcerer, breaks down each proof step-by-step, making the logic clear and easy to follow. Many found the focus on specific techniques like Double Inclusion particularly helpful. The short video format is seen as a positive, making it easy to fit study into a busy schedule. However, a significant number of students note a weakness: the course primarily involves watching proofs and offers very few practice problems, which some found made it hard to fully develop their own proof-writing skills. It's considered a solid supplement to other materials but perhaps less ideal as a standalone resource for absolute beginners to proofs who need extensive hands-on practice.

Course structure is primarily watching proofs being done.

"The best way to learn to write proofs is to watch someone else give careful proofs..."

"it's mostly just watching the instructor do proofs. While he explains his steps, it feels very passive."

"It's just video after video of the instructor writing proofs on a board. There's minimal interaction or exercises."

Course uses many short videos, good for daily study.

"The short video format made it easy to fit into my busy schedule."

"The pace is manageable thanks to the short videos."

"The short video format is convenient."

Valuable resource for supplementing other materials.

"Solid course on set theory proofs... The course covers the syllabus topics well."

"I was struggling with proofs in my university course, and this filled the gaps."

"A solid supplement to a textbook or a university course..."

Instructor clearly breaks down proofs step-by-step.

"The Math Sorcerer breaks down each proof step-by-step, explaining the logic clearly."

"His delivery is clear and straightforward, no unnecessary jargon."

"The instructor's calm and methodical approach is perfect. He takes his time with each step."

"A valuable resource for seeing correctly written proofs. The instructor is clear and the proofs are accurate."

Coverage of logical foundation is limited for beginners.

"The explanation of logic wasn't quite enough for me either; I felt lost without reviewing logic separately."

"The logical foundations section wasn't detailed enough; I had to constantly refer to other resources."

"Definitely not for beginners to proofs."

Course lacks sufficient practice problems for students.

"I wish there were more practice problems or exercises provided besides just watching the proofs."

"it's mostly just watching the instructor do proofs... didn't really help me *write* proofs myself as much as I hoped. I needed more active exercises."

"I struggled to apply what I saw to my own problems. The logical foundations section wasn't detailed enough..."

"My main critique is the lack of practice problems or challenges for the student to attempt before seeing the solution."

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in How to Write Proofs in Set Theory with the Math Sorcerer with these activities:

Review Mathematical Logic

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Strengthen your understanding of mathematical logic, a crucial foundation for writing proofs in set theory.

Browse courses on Mathematical Logic

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Review the definitions of logical connectives (AND, OR, NOT, IMPLIES, IFF).
Practice translating English statements into logical notation.
Work through examples of truth tables and logical equivalences.

Read 'How to Prove It' by Daniel Velleman

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Learn fundamental proof techniques and strategies to improve your ability to construct rigorous arguments in set theory.

View How to Prove It: A Structured Approach on Amazon

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Read the chapters on propositional logic and quantifiers.
Work through the examples and exercises in the book.
Apply the techniques learned to set theory problems.

Practice Set Theory Proofs

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Reinforce your understanding of set theory by working through a variety of proof exercises.

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Find a collection of set theory problems online or in a textbook.
Attempt to solve each problem on your own, writing out a complete proof.
Compare your solutions to the provided answers or seek feedback from peers.

Three other activities

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Show all six activities

Create a Proof Wiki

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Solidify your understanding by creating a wiki page that explains and demonstrates various proof techniques in set theory.

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Choose a specific proof technique (e.g., direct proof, proof by contradiction).
Write a clear and concise explanation of the technique.
Provide several examples of how to apply the technique to set theory problems.
Publish your wiki page and share it with other students.

Read 'Naive Set Theory' by Paul Halmos

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Gain a deeper understanding of the fundamental concepts and theorems of set theory.

View Naive Set Theory on Amazon

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Read the chapters on basic set operations and relations.
Work through the examples and exercises in the book.
Reflect on the philosophical implications of set theory.

Tutor Other Students

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Reinforce your knowledge by helping other students who are struggling with set theory proofs.

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Offer to tutor classmates or students in lower-level courses.
Explain concepts in different ways to cater to different learning styles.
Provide constructive feedback on their proof-writing attempts.

Career center

Learners who complete How to Write Proofs in Set Theory with the Math Sorcerer will develop knowledge and skills that may be useful to these careers:

Mathematician

Mathematicians conduct research in pure and applied mathematics, developing new theories and solving problems. This course directly aligns with the skillset needed to be a great mathematician. The core of mathematics lies in the ability to construct and understand rigorous proofs, which is the central focus of this course. The course's coverage of set theory, mathematical notation, and various proof techniques provides essential tools for mathematical reasoning. A mathematician should take this course for its intensive practice in proof writing, which is fundamental to mathematical research and problem-solving. This course can also help those seeking an advanced degree in mathematics.

See salaries and explore the career path for Mathematician

Software Verification Engineer

Software verification engineers ensure the correctness and reliability of software through rigorous testing and formal methods. A strong understanding of mathematical proofs is essential for applying formal verification techniques. This course directly contributes to success in this role, because the course’s focus on set theory and proof techniques provides the necessary background for formal verification. The software verification engineer benefits even further from the course's emphasis on mathematical notation. A software verification engineer should take this course for its intensive practice in proof writing, which is fundamental to validating software.

See salaries and explore the career path for Software Verification Engineer

Theoretical Computer Scientist

Theoretical computer scientists explore the fundamental principles of computation and information. This course can greatly help one move into this field. Proofs are a cornerstone of theoretical computer science, used to establish the correctness and efficiency of algorithms and data structures. The course’s explicit focus on constructing and understanding proofs in set theory builds a useful base for formal verification and complexity analysis. The course's coverage of mathematical notation, definitions of sets, and various proof techniques provides valuable tools for rigorous reasoning about computational systems. A theoretical computer scientist should take this course for its emphasis on the practice of proof writing, which is the bedrock of the discipline.

See salaries and explore the career path for Theoretical Computer Scientist

Database Theorist

Database theorists research the mathematical foundations of database systems. This course can be very useful to one who wishes to advance in this field. Set theory is fundamental to database theory, providing the basis for relational algebra and data modeling. The course’s explicit focus on proofs, especially those involving showing equivalence using methods like double inclusion, builds a strong base for work in database foundations. A database theorist benefits from the course's detailed coverage of mathematical notation and definitions of sets. A database theorist should take this course for its emphasis on the practice of proof writing, which is essential to the discipline.

See salaries and explore the career path for Database Theorist

Algorithm Designer

Algorithm designers create and optimize algorithms for solving computational problems. This course helps build a foundation for success in this area, because a deep understanding of mathematical proofs is essential for verifying the correctness and efficiency of algorithms. Set theory provides a framework for reasoning about collections of data and their relationships, which is fundamental to algorithm design. This course, with its focus on mathematical notation, set operations, and proof techniques, may be of tremendous help. It can also help the algorithm designer verify the correctness of their creations.

See salaries and explore the career path for Algorithm Designer

Math Teacher

Math teachers educate students in mathematical concepts and problem-solving skills. This course directly helps those who wish to grow in this field. A deep understanding of the subject matter is essential for effective teaching. With a deeper understanding of mathematics, the teacher may choose better and more insightful ways of explaining a topic to their students. This course, with its focus on set theory and proof techniques, builds a foundation for mathematical reasoning. A math teacher should take this course for its intensive practice in proof writing, which can be useful for preparing lessons and answering student questions. Math is often a difficult topic for younger learners, and a teacher who understands the concepts with greater clarity helps students succeed.

See salaries and explore the career path for Math Teacher

Data Scientist

A data scientist analyzes and interprets complex data sets, often using statistical methods and machine learning algorithms. This course may be useful as it can help build a foundation for understanding the mathematical underpinnings of these techniques. While data science is heavily applied, a solid grasp of set theory can aid in data manipulation, feature engineering, and the development of new algorithms. The course’s coverage of mathematical notation and set operations, combined with its emphasis on rigorous proofs, builds analytical skills applicable to data analysis. Success as a data scientist requires an understanding of the math behind the models.

See salaries and explore the career path for Data Scientist

Operations Research Analyst

An operations research analyst uses mathematical and analytical methods to help organizations make better decisions. Optimization problems often involve sets of constraints and objectives, and a solid understanding of set theory can be helpful in formulating and solving these problems. This course may be useful because the course's coverage of mathematical notation, set operations, and proof techniques builds the analytical skills needed for operations research. The analyst will feel more prepared after taking this course. This course is useful as it goes over fundamental methods of thinking and forming conclusions.

See salaries and explore the career path for Operations Research Analyst

Software Engineer

Software engineers design, develop, and maintain software systems. This course may be useful, as the ability to write proofs is valuable in ensuring software correctness and reliability. Formal methods, which rely on mathematical proofs to verify software properties, are increasingly used in critical applications. The course’s focus on set theory and proof techniques can aid in understanding and applying formal methods. Furthermore, the course's focus on rigourous detail helps to avoid unexpected behavior in software. A software engineer with a strong understanding of mathematical proofs helps build more trustworthy systems.

See salaries and explore the career path for Software Engineer

Cryptographer

A cryptographer works to secure data and communications through the use of encryption and decryption techniques. This course may be useful because it emphasizes rigorous proof construction, a skill central to verifying the security of cryptographic systems. The course's focus on understanding mathematical notation and set theory provides a foundation for the advanced mathematical concepts used in cryptography, such as modular arithmetic and finite fields. The detailed focus on proofs, especially those involving showing equivalence using methods like double inclusion, helps build the analytical rigor needed to break and build secure systems. As a cryptographer, a deep and provable understanding of data structures is critical, and this course can help.

See salaries and explore the career path for Cryptographer

Statistician

Statisticians collect, analyze, and interpret data to draw conclusions and make predictions. While statistics relies heavily on probability theory, a foundation in set theory can be valuable for understanding sample spaces and event algebras. This course may be useful as its coverage of mathematical notation, set operations, and proof techniques can improve abilities in rigorous analysis and logical reasoning. The course’s heavy emphasis on proofs in set theory can develop precision in thought, benefitting the statistician. A statistician with an understanding of mathematical proofs can derive better insights from data.

See salaries and explore the career path for Statistician

Data Analyst

Data analysts examine data to identify trends, patterns, and insights that can inform business decisions. Although data analysis is often more applied than theoretical, a solid grounding in mathematical logic and set theory can aid in data manipulation and interpretation. This course may be useful because it deepens one's analytical abilities for a more impactful career as a data analyst. The course's coverage of mathematical notation, set operations, and proof techniques can enhance skills in rigorous reasoning and problem-solving. A data analyst who has a deep understanding of mathematical proofs is of greater value.

See salaries and explore the career path for Data Analyst

Curriculum Developer

Curriculum developers design and create educational materials and courses. This may be for any subject, but in some cases, it may involve the design of a computer science or mathematics curriculum. This course may be useful because it provides insights into effective methods for teaching mathematical concepts, particularly those related to set theory and proof writing. The course's structure, short videos, and emphasis on practice may inform the design of engaging and effective learning experiences. This course provides the curriculum developer with an insight into this discipline.

See salaries and explore the career path for Curriculum Developer

Financial Modeler

Financial modelers create mathematical models to forecast financial performance and assess risk. While financial modeling relies heavily on statistical and econometric techniques, a strong foundation in mathematical reasoning can be valuable for understanding the underlying assumptions and limitations of these models. This course may be useful because it helps build a foundation for understanding the mathematical underpinnings of these models. With the knowledge of mathematical proofs taught by this course, the financial modeler is able to derive greater insights from their work.

See salaries and explore the career path for Financial Modeler

Technical Writer

Technical writers create documentation and guides for software, hardware, and other technical products. This course may be useful because it can enhance the ability to explain complex mathematical concepts clearly and concisely. The course’s focus on set theory and proof techniques can provide a deeper understanding of the subject matter, enabling the technical writer to produce more accurate and informative documentation. The writer will also be able to follow the course's teaching methods and incorporate them into their own writing. The technical writer will write with greater authority.

See salaries and explore the career path for Technical Writer

How to Write Proofs in Set Theory with the Math Sorcerer

What's inside

Learning objectives

Syllabus

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Reviews summary

Learning to write set theory proofs

Activities

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