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How to Prove It – Solutions
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- Summary of article content: Articles about How to Prove It – Solutions Chapter – 1, Sentential Logic. Section – 1.3 – Variable and Sets. July 18, 2015. Summary. Statements with Variables. For eg: “x is divisible by 9”, … …
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reference request – Solution manual for ” How to Prove it by D. Vellemen ” – Mathematics Stack Exchange
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GitHub – psibi/how-to-prove: My Solution to Velleman’s book
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How prove it structured approach 3rd edition | Logic, categories and sets | Cambridge University Press
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- Summary of article content: Articles about How prove it structured approach 3rd edition | Logic, categories and sets | Cambridge University Press How to Prove It A Structured Approach. 3rd Edition. $30.00 ( ) USD. Author: Daniel J. Velleman, Amherst College, Massachusetts. …
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how to prove it a structured approach solutions
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- Summary of article content: Articles about how to prove it a structured approach solutions computer scientists to embrace the structured approach to programming ap- … complete solution; in some cases, it is a sketch of a solution, or a hint. …
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How To Prove It: A Structured Approach Third Edition Solutions Manual – PDF Free Download
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How to Prove It – Solutions
Summary
Statements with Variables. For eg: “x is divisible by 9”, “y is a person” are statements. Here x, and y are variables.
These statements are true or false based on the value of variables.
Sets, a collection of objects.
Bound and Unbound variables. Eg: \( y ∈ {x\,\vert\,x^3 < 9} \) , $y$ is a free variable, whereas $x$ is a bound variable. is a free variable, whereas is a bound variable. Free variables in a statement are for objects for which statement is talking about. Bound variables are just dummy variables to help express the idea. Thus bound variables dont represent any object of the statement. The truth set of a statement P(x) is the set of all values of x that make the statement P(x) true. The set of all possible values of variables is call universe of discourse. Or variables range over this universe. In general, $y ∈ \{x ∈ A\,\vert\,P(x)\}$ means the same thing as $y ∈ A ∧ P(y)$ . Solutions Soln1 (a) $D(6,3) \land D(9,3) \land D(15, 3)$ where $D(x, y)$ means $x$ is divisible by $y$. (b) $D(x,2) \land D(x,3) \land \lnot D(x, 4)$ where $D(x, y)$ means $x$ is divisible by $y$. (c) $(\lnot P(x) \land P(y)) \lor (P(x) \land \lnot P(y)$ where $P(x) = \{ x \in \mathbb{N}\,\vert\, x \text{ is prime} \}$. Soln2 (a) $M(x) \land M(y) \land (T(x,y) \lor T(y,x))$ where $M(x)$ is “x is men”, $T(x, y)$ means “x is taller than y”. (b) $[(B(x) \lor B(y)) \land (R(x) \lor R(y)]$ where $B(x)\text{ and }R(x)$ means “x has brown eyes” and “x has brown hairs” respectively. (c) $[(B(x) \land R(x)) \lor (B(y) \land R(y)]$ where $B(x)\text{ and }R(x)$ means “x has brown eyes” and “x has brown hairs” respectively. Soln3 (a) $\{ x\,\vert\,x\text{ is a planet }\}$ (b) $\{ x\,\vert\,x\text{ is a university }\}$ (c) $\{ x\,\vert\,x\text{ is a state in US }\}$ (d) $\{ x\,\vert\,x\text{ is a province in Canada }\}$ Soln4 (a) \[ { x^2\,\vert\, x > 0 \text{ and } x \in \mathbb{N} } \]
(b) $\{ 2^x\,\vert\, x \in \mathbb{N} \}$
(c) $\{ x \in \mathbb{N}\,\vert\, 10 \le x \le 19 \}$
Soln5
(a) $−3 ∈ \{x ∈ \mathbb{R}\vert\,13 − 2x > 1\} \Rightarrow -3 \in \mathbb{R} \land 19 > 1$. No free variables in the statement. Statement is true.
(b) $4 ∈ \{x ∈ \mathbb{R^+}\vert\,13 − 2x > 1\} \Rightarrow 4 \in \mathbb{R^+} \land 5 > 1$. No free variables in the statement. Statement is false.
(c) $5
otin \{x ∈ \mathbb{R}\vert\,13 − 2x > c\} \Rightarrow \lnot{ \{ 5 \in \mathbb{R} \land 3 > c \}} \Rightarrow 5
otin \mathbb{R} \lor 3 \le c$. One free variable(c) in the statement. (Thanks Maxwell for the correction)
Soln6
(a) $(w ∈ \mathbb{R}) \land (13 – 2w > c)$. There are two free variables $w$ and $c$.
(b) $(4 \in \mathbb{R}) \land (13 – 2 \times 4 \in P) \Rightarrow (4 \in \mathbb{R}) \land (5 \in P)$. The statement has no free variables. It is a true statement.
(c) $(4 \in P) \land (13 – 2 \times 4 > 1) \Rightarrow (4 \in P) \land (5 > 1)$. The statement has no free variables. It is a false statement.
Soln7
(a) {Conrad Hilton Jr., Michael Wilding, Michael Todd, Eddie Fisher, Richard Burton, John Warner, Larry Fortensky}.
(b) $\{ \lor, \land, \lnot \}$
(c) { Daniel Velleman }
Soln8
(a) {1, 3}
(b) $\phi$
(c)
Update:
As pointed out in comments, I got this wrong first time. Here is the correct answer:
$\{ x ∈ R \,\vert\, x^2 < 25 \}$ or, equivalently $\,\vert x \vert\, \lt 5$. Old Answer: $$ \{1, 2, 3, 4, 5, 6, 7 \} $$
psibi/how-to-prove: My Solution to Velleman’s book
How to Prove It: A Structured Approach
Contains solution for the Velleman’s book.
The reason I have started studying this is to ultimately study type theory.
Feel free to raise issue if you think a proof is wrong or if it needs some clarification. Pull requests and contributions are welcome.
Credits
David Cantrell for Section 3.3 and various other improvements.
From Chapter 2, I have moved from markdown format to Latex as it helps in much easier rendering of mathematical symbols. It can be compiled using pdflatex :
pdflatex tex_filename
Although I did solve Chapter 3 problems, I haven’t uploaded all of them yet because of my laziness.
How prove it structured approach 3rd edition
Proofs play a central role in advanced mathematics and theoretical computer science, yet many students struggle the first time they take a course in which proofs play a significant role. This bestselling text’s third edition helps students transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. Featuring over 150 new exercises and a new chapter on number theory, this new edition introduces students to the world of advanced mathematics through the mastery of proofs. The book begins with the basic concepts of logic and set theory to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for an analysis of techniques that can be used to build up complex proofs step by step, using detailed ‘scratch work’ sections to expose the machinery of proofs about numbers, sets, relations, and functions. Assuming no background beyond standard high school mathematics, this book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and, of course, mathematicians.
Perfect for self-study, an introduction to proofs course, or as a supplementary text for a discrete mathematics course or foundations of computing course
Systematic and thorough, showing how several techniques can be combined to construct a complex proof
Covers logic, set theory, relations, functions, and cardinality
Reviews & endorsements
‘Not only does this book help students learn how to prove results, it highlights why we care so much. It starts in the introduction with some simple conjectures and gathering data, quickly disproving the first but amassing support for the second. Will that pattern persist? How can these observations lead us to a proof? The book is engagingly written, and covers – in clear and great detail – many proof techniques. There is a wealth of good exercises at various levels. I’ve taught problem solving before (at The Ohio State University and Williams College), and this book has been a great addition to the resources I recommend to my students.’ Steven J. Miller, Williams College, Massachusetts
‘This book is my go-to resource for students struggling with how to write mathematical proofs. Beyond its plentiful examples, Velleman clearly lays out the techniques and principles so often glossed over in other texts.’ Rafael Frongillo, University of Colorado, Boulder ‘I’ve been using this book religiously for the last eight years. It builds a strong foundation in proof writing and creates the axiomatic framework for future higher-level mathematics courses. Even when teaching more advanced courses, I recommend students to read chapter 3 (Proofs) since it is, in my opinion, the best written exposition of proof writing techniques and strategies. This third edition brings a new chapter (Number Theory), which gives the instructor a few more topics to choose from when teaching a fundamental course in mathematics. I will keep using it and recommending it to everyone, professors and students alike.’ Mihai Bailesteanu, Central Connecticut State University ‘Professor Velleman sets himself the difficult task of bridging the gap between algorithmic and proof-based mathematics. By focusing on the basic ideas, he succeeded admirably. Many similar books are available, but none are more treasured by beginning students. In the Third Edition, the constant pursuit of excellence is further reinforced.’ Taje Ramsamujh, Florida International University ‘Proofs are central to mathematical development. They are the tools used by mathematicians to establish and communicate their results. The developing mathematician often learns what constitutes a proof and how to present it by osmosis. How to Prove It aims at changing that. It offers a systematic introduction to the development, structuring, and presentation of logical mathematical arguments, i.e. proofs. The approach is based on the language of first-order logic and supported by proof techniques in the style of natural deduction. The art of proving is exercised with naive set theory and elementary number theory throughout the book. As such, it will prove invaluable to first-year undergraduate students in mathematics and computer science.’ Marcelo Fiore, University of Cambridge ‘Overall, this is an engagingly-written and effective book for illuminating thinking about and building a careful foundation in proof techniques. I could see it working in an introduction to proof course or a course introducing discrete mathematics topics alongside proof techniques. As a self-study guide, I could see it working as it so well engages the reader, depending on how able they are to navigate the cultural context in some examples.’ Peter Rowlett, LMS Newsletter
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Edition: 3rd Edition
Date Published: July 2019
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