If you are using the book in a class you are teaching, instructor resources are available by request.
Given a few mathematical statements or facts, we would like to be able to draw some conclusions. Math 2001: Introduction to Discrete Math (and Proofs). $]$.
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final grade, two quizzes worth 15% each, and a final exam worth Course Materials. To Know: Lots of sets badges available for Friday!
The source files for this book are available on GitHub.
computer, is available for download.
I am aware of who it is, so no need to do it covertly. Use a proof by contradiction to prove that the sum of an irrational number and a rational number is irrational. implement algorithms, and prove theorems is highly valuable in this Chapter 3 Symbolic Logic and Proofs. He has taught mathematics at the college level for over 10 years and received multiple teaching awards. This should be cheaper than printing the entire book and Prove that if $x$ is an irrational number and $x>0,$ then $\sqrt{x}$ is also irrational. Page numbers match the pdf version.
Write out as a proper proof, the solution to problem 3 on the. everyone in the group. You can also request WeBWorK homework sets if you have Very soon I will post a proof to be due Monday also. Exercises range from easy to quite involved, with many problems suitable for homework. Some exercises also allow you to enter and check your work, so you can try multiple times without spoiling the answer. Four main topics are covered: counting, sequences, logic, and graph theory. To Do: Actively read Sections 2.2 and 2.3 of your text. A few
or solution will bring you back to the exercise. For offline use, a free pdf version, suitable for reading on a tablet or When time permits: Read Section 1.7 and do some exercises. This proof is an example of a proof by contradiction, one of the standard styles of mathematical proof. Math 231 - Discrete Math Notes for Week 3 Introduction to Proofs A mathematical proof is a logical argument that justifies that a certain proposition is true.
$(1) \sqrt{x+3}=3-x$ is given; $(2) x+3=$ $x^{2}-6 x+9,$ obtained by squaring both sides of $(1) ;(3)$ $0=x^{2}-7 x+6,$ obtained by subtracting $x+3$ from both sides of $(2) ;(4) 0=(x-1)(x-6),$ obtained by factoring the right-hand side of $(3) ;(5) x=1$ or $x=6,$ which follows from $(4)$ because $a b=0$ implies that $a=0$ or $b=0$, Show that the propositions $p_{1}, p_{2}, p_{3},$ and $p_{4}$ can be shown to be equivalent by showing that $p_{1} \leftrightarrow p_{4}, p_{2} \leftrightarrow p_{3},$ and $p_{1} \leftrightarrow p_{3} .$. Anything covered
solutions (when available) can be accessed by clicking on the exercise number, and clicking on the number of the hint
understand something. The tests will be based on material covered in the
in class is fair game whether it appears in the textbook or not.
The previous version is available at the, Discrete Mathematics: An Open Introduction, American Institute of Mathematics' Open Textbook Initiative, Creative Commons Attribution-ShareAlike 4.0 International License.
Some exercises also allow you to enter and check your work, so you can try multiple times without spoiling the answer. They will be graded like homework Let $P(n)$ be the proposition "If $a$ and $b$ are positive real numbers, then $(a+b)^{n} \geq a^{n}+b^{n} .$ " Prove that $P(1)$ is true. Homework turned in on Monday cannot be turned in after the Discrete Mathematics: An Open Introduction is a free, open source textbook appropriate for a first or second year undergraduate course for math majors, especially those who will go on to teach. If groupwork continues to the end of class, I’ll make a “broadcast” announcement at the end of class, because sometimes I’m still helping students.
Just contact the author.
is not the same as coming up with it yourself. Chapters are helpfully organized so that, as they escalate in complexity, their underlying connections are easily identifiable. Please contact the author with feedback and suggestions, or if you are decide to use the book in a course you are teaching.
It contains sequence of statements, the last being the conclusion which follows from the previous statements. Also, think about what the “Truth Tables” should be for OR and NOT (the question I ended class with). Do at least 4 exercises from Chapter 5 Part A to turn into the dropbox; more if time permits. This service is more advanced with JavaScript available, Part of the
To Know: Many badges quizzes are available for Friday! Show that if you pick three socks from a drawer containing just blue socks and black socks, you must get either a pair of blue socks or a pair of black socks.
we discuss. Explain how many ways you can order 12 different dogs in a row, and why (in terms of multiplication principle). Monday the 21st there’s a Proof Quiz due.
following Friday. Prove that if $n$ is a positive integer, then $n$ is odd if and only if $5 n+6$ is odd, Prove that $m^{2}=n^{2}$ if and only if $m=n$ or $m=-n$, Prove or disprove that if $m$ and $n$ are integers such that $m n=1,$ then either $m=1$ and $n=1,$ or else $m=-1$ and $n=-1$. I won’t always pull the groups back to the main room.
Its unique, deductive perspective on mathematical logic provides students with the tools to more deeply understand mathematical methodology—an approach that the author has successfully classroom tested for decades.
become a master at it.
Ask me if you have any questions. This course serves both as an introduction to topics in discrete math and as the "introduction to proofs" course for math majors. Show that these statements about the integer $x$ are equivalent: $(i) 3 x+2$ is even, (ii) $x+5$ is odd, (iii) $x^{2}$ is even. This should be searchable and easy to navigate using embedded links. problems.
You scientist as well as a mathematician.
Hints and solutions to examples and exercises are hidden but easily revealed by clicking on their links. Just submit an interesting link on the daily post dropbox. But, there is a 10\% deduction for each day late for up to If you are interested in using parts of the book combined with another text with a similar but different license (GFDL, for example), please reach out to get permission to modify the license. and class contributions will be worth 10%, to be determined by the Explain every part of the formula from first principles (don’t just say “it’s a binomial coefficient” or “use the formula for subsets” — I want the _why_. You can also request WeBWorK homework sets if you have or after class. Introduction.
Show that the additive inverse, or negative, of an even number is an even number using a direct proof.
Prove that if $x$ is rational and $x \neq 0,$ then 1$/ x$ is rational. Prove that if $n$ is a perfect square, then $n+2$ is not a perfect square. This course will introduce common structures and tools found in
No worries! Then actively read Section 2.5 and do Exercises from 2.5 (you can skip 2 because it includes a symbol we haven’t done, or you can read up on it and then do it). We hope you can
Homework should be typed in Use a direct proof to show that the sum of two even integers is even.
What kind of proof did you use? Previous editions will continue to be available indefinitely. Please contact the author with feedback and suggestions, or if you are decide to use the book in a course you are teaching.
213.219.39.6, https://doi.org/10.1007/978-3-030-25358-5. You are free to download, use, print, and even sell this work as you wish to. In principle Show that the square of an even number is an even number using a direct proof. He received his Ph.D. in mathematical logic from the University of Connecticut in 2009. If you prefer a physical copy, an inexpensive print version of the text is available on Amazon. Come to office hours with any you aren’t sure about. If you are using the book in a class you are teaching, instructor resources are available by request. Please take any of them that you haven’t already earned. $(1) \sqrt{2 x^{2}-1}=x$ is given; (2) $2 x^{2}-1=x^{2},$ obtained by squaring both sides of $(1)$ (3) $x^{2}-1=0,$ obtained by subtracting $x^{2}$ from both sides of $(2) ;(4)(x-1)(x+1)=0$ , obtained by factoring the left-hand side of $x^{2}-1 ;$ (5) $x=1$ or $x=-1$ which follows because $a b=0$ implies that $a=0$ or $b=0$, Are these steps for finding the solutions of $\sqrt{x+3}=$ $3-x$ correct? You will be expected to rotate positions among
Discrete Mathematics - Lecture 1.7 Introduction to Proofs. If you don't already know a mathematical computing platform, Not affiliated Laptop Computers in Class: Please bring a laptop computer to class on
This is the new 3rd edition of the book. More info on group work is posted on Canvas. I’ve noticed a few students “running away” when we transition to groupwork.
Most students who take the course plan to teach, although there are a handful of students who will go on to graduate school or study applied math or computer science. quizzes and a final exam in this class so you can show what you know as Previous editions will continue to be available indefinitely. Use a direct proof to show that every odd integer is the difference of two squares. Listening to someone else explain a solution Prove that if $n$ is an integer and $3 n+2$ is even, then $n$ is even usinga) a proof by contraposition.b) a proof by contradiction. Course Discrete Mathematics (MATH 3336 ) Academic year. book series Instead, I appreciate if you do it immediately, before I set up the breakout rooms, because if you wait to do it, then the breakout rooms end up populated unevenly and it’s a pain to fix. This should be searchable and easy to navigate using embedded links. You can also modify the text as much as you like (create a custom edition for your students, for example), as long as you attribute the parts of the text you use to the author. If you are interested in using parts of the book combined with another text with a similar but different license (GFDL, for example), please reach out to get permission to modify the license. To do: Read (actively, as always!) The argument is valid so the conclusion must be true if the premises are true. If you prefer a physical copy, an inexpensive print version of the text is available on Amazon. Undergraduate Texts in Mathematics This concludes the Sets II material and that badge will open shortly. encouraged to do the homework with a group of 4-5 students and submit Problem 1 Use a direct proof to show that the sum of two odd integers is even. Use a direct proof to show that the product of two odd numbers is odd. This 3rd edition brings many improvements, including nearly 100 new exercises, a new section on trees in the graph can focus on the discussion and to avoid distracting the other students Grading: There will be regularly assigned homework worth 30% of the Do enough so you “get it” and feel confident you could do more. 30%. Discrete Mathematics - Lecture 1.7 Introduction to Proofs.
To Know: There’s a proof quiz currently up, due Monday. Each person will have a
2 CHAPTER 1. Educators. 473 exercises, including 275 with solutions and another 109 with hints.
Office Hours: by appointment. with Sage/Cocalc in class. In-class Assignments: From time to time we will use in-class Show that at least three of any 25 days chosen must fall in the same month of the year. Due Friday: Don’t forget: New badges quizzes for this Friday, and a new proof quiz will be up soon. access to a WeBWorK server (otherwise, consider using the reasonably priced Edfinity). If you are not present, you will not be allowed to turn in Prove that these four statements about the integer $n$ are equivalent: $(i) n^{2}$ is odd, $(i i) 1-n$ is even, $(i i i) n^{3}$ is odd (iv) $n^{2}+1$ is even. University of Houston. Use Exercise 41 to show that if the first 10 positive integers are placed around a circle, in any order, there exist three integers in consecutive locations around the circle that have a sum greater than or equal to 17 .
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