Introduction to advanced mathematics MATH 215 Filippo Calderoni - - PowerPoint PPT Presentation

Introduction to advanced mathematics

MATH – 215 Filippo Calderoni

University of Illinois at Chicago

August 26, 2019

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Introducing myself

The instructor

§ Research Assistant Professor in the

department of Mathematics Statistics and Computer Science

§ PhD in 2018 at University of Turin

(Torino, Italy)

§ Specialize in Mathematical Logic

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General Info

Room: AH 303 Schedule: Mon–Wed–Fri at 09:00 – 09:50 Office hours: Friday 10–12 and by appointment. Email: fcaldero@uic.edu Course webpage: click here!

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General Info: textbook

Main textbook: Eccles, P. An Introduction to Mathematical Reasoning: Numbers, Sets and Functions. Cambridge: Cambridge University Press. (1997) Another textbook: Sundstrom, T. Mathematical reasoning: writing and proof (Available

• nline!)

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Important dates

Date Event August 26, M First class. September 2, M Labor Day holiday. No classes. October 4, F Midterm I. November 4, M Midterm II. November 29, F Thanksgiving holiday. No classes. December 6, F Instruction ends. December 10, Tu Final exam. (10:30–12:30)

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Participation

Attendance

§ Attendance is not required, but it is strongly recommended. § Students are responsible for all of the material covered in the lectures. § Find a concise summary of each lecture in course webpage. § If you miss a lecture, you should ask your classmates for notes.

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Participation

Attendance

§ Attendance is not required, but it is strongly recommended. § Students are responsible for all of the material covered in the lectures. § Find a concise summary of each lecture in course webpage. § If you miss a lecture, you should ask your classmates for notes.

Instructor’s tip Problems on the exams will be similar to problems discussed in class.

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Participation

Attendance

§ Attendance is not required, but it is strongly recommended. § Students are responsible for all of the material covered in the lectures. § Find a concise summary of each lecture in course webpage. § If you miss a lecture, you should ask your classmates for notes.

Instructor’s tip Problems on the exams will be similar to problems discussed in class. You are encouraged to participate actively: Ask questions, make comments, stop me.

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Active participation

Actively participation consists of various activity:

§ coming to office hours,

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Active participation

Actively participation consists of various activity:

§ coming to office hours, § exchanging email communication with the instructor,

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Active participation

Actively participation consists of various activity:

§ coming to office hours, § exchanging email communication with the instructor, § participating to in-class discussion,

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Active participation

Actively participation consists of various activity:

§ coming to office hours, § exchanging email communication with the instructor, § participating to in-class discussion, § . . .

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Active participation

Actively participation consists of various activity:

§ coming to office hours, § exchanging email communication with the instructor, § participating to in-class discussion, § . . .

Ask questions, make comments, stop me.

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Grades

The final grade for the course will be based on the grade of the following components: homework assignments, two in-class midterm exams, and a final exam. The final grade Each component will be weighted as follows:

§ 20% Homework (lowest score will be dropped) § 20% Midterm I (Friday October 4, in class) § 20% Midterm II (Monday November 4, in class) § 40% Final Exam

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Homework

§ The usual schedule will be one problem set per week. § Problem sets will be posted on the course webpage, typically on

Friday.

§ Usually the due date will be on a class day, in which case the set is

due at the beginning of class.

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Homework

§ The usual schedule will be one problem set per week. § Problem sets will be posted on the course webpage, typically on

Friday.

§ Usually the due date will be on a class day, in which case the set is

due at the beginning of class. First week grace No homework on first week!!!

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Homework submission

Paper submission is required. Submission guideline

§ Use full-sized paper (A4, notebook paper, etc)

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Homework submission

Paper submission is required. Submission guideline

§ Use full-sized paper (A4, notebook paper, etc) § Write your full name near the top of the front page.

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Homework submission

Paper submission is required. Submission guideline

§ Use full-sized paper (A4, notebook paper, etc) § Write your full name near the top of the front page. § Please, staple multiple pages.

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Homework submission

Paper submission is required. Submission guideline

§ Use full-sized paper (A4, notebook paper, etc) § Write your full name near the top of the front page. § Please, staple multiple pages. § Typesetting solutions is not required, but...

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Homework submission

Paper submission is required. Submission guideline

§ Use full-sized paper (A4, notebook paper, etc) § Write your full name near the top of the front page. § Please, staple multiple pages. § Typesetting solutions is not required, but... § ...writing should be legible!

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Homework submission

Paper submission is required. Submission guideline

§ Use full-sized paper (A4, notebook paper, etc) § Write your full name near the top of the front page. § Please, staple multiple pages. § Typesetting solutions is not required, but... § ...writing should be legible!

Late submission Late homework will not be accepted without prior permission from the instructor.

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Homework submission

§ Homework is individual § Solutions must be written in your own words!

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Homework submission

§ Homework is individual § Solutions must be written in your own words!

Instructor’s tip This course addresses an essential aspect of math: writing down proofs. Don’t overlook exposition!

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Homework submission

§ Homework is individual § Solutions must be written in your own words!

Instructor’s tip This course addresses an essential aspect of math: writing down proofs. Don’t overlook exposition! Ask questions, make comments, stop me.

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Exams

General policy

§ Notes or books will not be allowed. § Collaboration of any kind on the exam is prohibited. § Makeup exams will only be given in case of a verifiable emergency or

a formal request by the UIC athletic department.

§ Do not schedule travel on an exam date.

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“The faculty of the University of Illinois at Chicago shall make every effort to avoid scheduling examinations or requiring that student projects be turned in or completed on religious holidays. Students who wish to

• bserve their religious holidays shall notify the faculty member by

the tenth day of the semester of the date when they will be absent unless the religious holiday is observed on or before the tenth day of the semester. In such cases, the students shall notify the faculty member at least five days in advance of the date when he/she will be absent. The faculty member shall make every reasonable effort to honor the request, not penalize the student for missing the class, and if an examination or project is due during the absence, give the student an exam or assignment equivalent to the one completed by those students in attendance. If the student feels aggrieved, he/she may request remedy through the campus grievance procedure.” Please, visit https://oae.uic.edu/religious-calendar/

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The course

Content Introduction to methods of proofs used in different fields in mathematics. Prerequisite(s): Grade of C or better in MATH 181 and approval of the department.

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Proof: “The process or an instance of establishing the validity of a statement especially by derivation from other statements in accordance with principles of reasoning.”1 In particular, a proof must be ... . (Fill in the gap)

§ complicated § convincing § correct § confusing § chalkboard written § lengthy § consistent § incomplete § readable

1From Webster dictionary. 15/20

Proof: “The process or an instance of establishing the validity of a statement especially by derivation from other statements in accordance with principles of reasoning.”1 In particular, a proof must be ... . (Fill in the gap)

§ complicated § convincing § correct § confusing § chalkboard written § lengthy § consistent § incomplete § readable

1From Webster dictionary. 15/20

The skill of writing proofs

§ Comparable to write an essay, coding, give precise instructions. § Can also be viewed as a pedagogical act.

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The skill of writing proofs

§ Comparable to write an essay, coding, give precise instructions. § Can also be viewed as a pedagogical act.

Consist of various aspects:

§ Understanding the underlying mathematical ideas. § Using the appropriate vocabulary/specific terms. § Motivating each of your assertions. § Avoiding wrong and misleading conclusions.

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The skill of writing proofs

§ Comparable to write an essay, coding, give precise instructions. § Can also be viewed as a pedagogical act.

Consist of various aspects:

§ Understanding the underlying mathematical ideas. § Using the appropriate vocabulary/specific terms. § Motivating each of your assertions. § Avoiding wrong and misleading conclusions.

Hasn’t much to do with:

§ Performing difficult computations.

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Avoiding wrong conclusions

Example 1 Fermat conjectured that all numbers of the form 22n ` 1 were prime, after having checked this for n ď 4...

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Avoiding wrong conclusions

Example 1 Fermat conjectured that all numbers of the form 22n ` 1 were prime, after having checked this for n ď 4... 220 “ 21 ` 1 “ 3

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Avoiding wrong conclusions

Example 1 Fermat conjectured that all numbers of the form 22n ` 1 were prime, after having checked this for n ď 4... 220 “ 21 ` 1 “ 3 221 “ 22 ` 1 “ 5

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Avoiding wrong conclusions

Example 1 Fermat conjectured that all numbers of the form 22n ` 1 were prime, after having checked this for n ď 4... 220 “ 21 ` 1 “ 3 221 “ 22 ` 1 “ 5 222 “ 24 ` 1 “ 17

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Avoiding wrong conclusions

Example 1 Fermat conjectured that all numbers of the form 22n ` 1 were prime, after having checked this for n ď 4... 220 “ 21 ` 1 “ 3 221 “ 22 ` 1 “ 5 222 “ 24 ` 1 “ 17 . . .

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Avoiding wrong conclusions

Example 1 Fermat conjectured that all numbers of the form 22n ` 1 were prime, after having checked this for n ď 4... 220 “ 21 ` 1 “ 3 221 “ 22 ` 1 “ 5 222 “ 24 ` 1 “ 17 . . . ...but Euler refuted this conjecture by showing that 225 ` 1 “ 4292967297 “ 641 ˆ 6700417.

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Avoiding wrong conclusions

Example 2 The property Ppnq asserting “n2 ´ 79 ˆ n ` 1601 is a prime number” is true for 1 ď n ă 80...

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Avoiding wrong conclusions

Example 2 The property Ppnq asserting “n2 ´ 79 ˆ n ` 1601 is a prime number” is true for 1 ď n ă 80... ...but false for n “ 80 since 802 ´ 79 ˆ 80 ` 1601 “ 1681 “ 412.

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In 1900, at the second International Mathematical Congress in Paris, Hilbert posed his (now famous) list of 23 questions to guide mathematics for the “coming century”

§ The problem of Continuum Hypothesis (CH) was the first problem

in the list.

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In 1900, at the second International Mathematical Congress in Paris, Hilbert posed his (now famous) list of 23 questions to guide mathematics for the “coming century”

§ The problem of Continuum Hypothesis (CH) was the first problem

in the list. ...4 years later In 1904, at the third International Mathematical Congress in Heidelberg, Konig gave a lecture in which he claimed to have solved CH.

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In 1900, at the second International Mathematical Congress in Paris, Hilbert posed his (now famous) list of 23 questions to guide mathematics for the “coming century”

§ The problem of Continuum Hypothesis (CH) was the first problem

in the list. ...4 years later In 1904, at the third International Mathematical Congress in Heidelberg, Konig gave a lecture in which he claimed to have solved CH.

§ All parallel session at the congress were cancelled so that everybody

could attend Konig’s lecture.

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In 1900, at the second International Mathematical Congress in Paris, Hilbert posed his (now famous) list of 23 questions to guide mathematics for the “coming century”

§ The problem of Continuum Hypothesis (CH) was the first problem

in the list. ...4 years later In 1904, at the third International Mathematical Congress in Heidelberg, Konig gave a lecture in which he claimed to have solved CH.

§ All parallel session at the congress were cancelled so that everybody

could attend Konig’s lecture.

§ The announcement was sensational and was widely reported by the

press.

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In 1900, at the second International Mathematical Congress in Paris, Hilbert posed his (now famous) list of 23 questions to guide mathematics for the “coming century”

§ The problem of Continuum Hypothesis (CH) was the first problem

in the list. ...4 years later In 1904, at the third International Mathematical Congress in Heidelberg, Konig gave a lecture in which he claimed to have solved CH.

§ All parallel session at the congress were cancelled so that everybody

could attend Konig’s lecture.

§ The announcement was sensational and was widely reported by the

press.

§ The Grand Duke Friedrich I of Prussia had Felix Klein explain the

entire matter to him personally.

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In 1900, at the second International Mathematical Congress in Paris, Hilbert posed his (now famous) list of 23 questions to guide mathematics for the “coming century”

§ The problem of Continuum Hypothesis (CH) was the first problem

in the list. ...4 years later In 1904, at the third International Mathematical Congress in Heidelberg, Konig gave a lecture in which he claimed to have solved CH.

§ All parallel session at the congress were cancelled so that everybody

could attend Konig’s lecture.

§ The announcement was sensational and was widely reported by the

press.

§ The Grand Duke Friedrich I of Prussia had Felix Klein explain the

entire matter to him personally.

§ ... the proof was wrong!

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Contents

Language of mathematics/set theory: sets, functions. Principle of induction, the binomial theorem, basic number theory. Transitive relations, order relations. Equivalence relations, modular arithmetic.

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Contents

Language of mathematics/set theory: sets, functions. Principle of induction, the binomial theorem, basic number theory. Transitive relations, order relations. Equivalence relations, modular arithmetic. Questions?

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