Summary
MENU General Guidelines for Technical Writing Mathematical Writing An Introduction to Formal Logic Exercises

1. General Guidelines for Technical Writing

1. Target (i.e., Anticipated Readers)
2. Theme (i.e., Primary Objective)
3. Content (i.e., What to Write)
4. Organization (i.e., How to Structure Text)
5. Style (i.e., How to Express in Writing)

Differences between Technical Writing and General Writing

(quotation from

Kavita Tyagi and Padma Misra, Basic Technical Communication, Revised Ed., p.9, PHI Learning, New Delhi, 2012)

	Technical Writing	General Writing
Document Type	All professional, scientific and specialist documents, drafts, reports, letters, papers, theses	Literary and other types of writing
Style	Familiar, simple, clear and precise and of evaluating value	Poetic, rhetorical or elegant and carries stamp of individuality
Skills	Acquired through practice	Creative and innovative with an inborn talent
Format	Strict and well defined so that the reader can understand the organization of document	No set pattern and predefined organization
Language	Simple, straightforward, objective, rational and scientific	Elegant or creative. Can be poetic, literary or generic
Words	Technical words and their explanation	Descriptive and literary composition
Content	Preplanned on the basis of information collected	Spontaneous and written on-the-spur of the moment
Profession	Pertains to profession	Relates to society in general
Function	Instructs, informs and persuades	Amuses, inspires and educates
Diction	It is simple and effective	May use complex and long sentences, but the meaning will be clear

• Manuals:
  User's Manuals, Installation Guides, Maintenance Manuals, Reference Manuals, etc.
• Specifications:
  Electrical Specifications, Physical Specifications, Design Specifications, Requirements Specifications, Functional Specifications, etc.
• Reports:
  Term Papers for Assignments, Solutions for Homework, Lab Reports, etc.
• Papers:
  Journal Papers, Conference Papers, Technical Reports, etc.
• Bachelor Theses, Master Theses and Ph.D. Dissertations

• Technical Writing Tutorial at MIT
• A Teacher's Guide to Technical Writing by Dr. Steven M. Gerson, Johnson County Community College
• UIUC TOEFL-Prep Writing Practice Site
• Walden University Writing Center - Composition -
• Avoid These Technical Writing Mistakes by Robert W. Bly, Chemical Engineering Progress, June 1998
• NASA's Handbook on Technical Writing
• Chicago Manual of Style (CMS)

• Writing Technical Articles by Dr. Henning Schulzrinne, Dept. of Computer Science, Columbia University
• Writing a Technical Paper by Dr. Michael D. Ernst, MIT
• How to Write a Paper by Dr. Armando Fox, UC Berkeley

Remarks on Common Mistakes by Students in Technical Writing

1. Do not use the first person singular pronoun "I" even if a paper was written by a single author.
   Similarly, do not use "my" and use "our" instead.
   It is also very unusual to use the second person singular/plural pronoun "you" in a technical article.

     Bad:    I present the time complexity of an algorithm in this paper.
     OK:     We present the time complexity of an algorithm in this paper.
     Better: The time complexity of an algorithm is presented in this paper.   [traditional style:  passive voice]
             This paper presents the time complexity of an algorithm.          [modern style: active voice]
   

   If the subject of a sentence is truly an author or authors, the first person plural pronoun "we" is OK.
   Otherwise, it is recommended to write the subject explicitly and avoid a pronoun unless what the pronoun refers to is definitely obvious to the readers,
   where the subject is an agent of the action (i.e., verb) mentioned in a sentence.

     Bad:    We first sort points in the nondecreasing order of their x-coordinate values.
             Then, we find a closest pair of points.
     Good:   An algorithm first sorts points in the nondecreasing order of their x-coordinate values.
             Then, the algorithm finds a closest pair of points.
   

2. Write sentences in the simple present tense unless another tense is definitely needed
   (e.g., in the last section "Conclusions", sentences summarizing a paper are written in the past tense).
   Examples:

     "This paper models a problem of finding
     shortest paths from a single source to
     all other vertices in a digraph."
   
     "This paper presents an algorithm for
     determining whether a given positive integer
     is prime."
   

   Examples of Exceptions:

     "In 1883, the puzzle called the Towers of Hanoi was invented
     by the French mathematician Édouard Lucas."
   
     "E. W. Dijkstra proposed a greedy algorithm
     for finding single-source shortest paths in
     1959."
   
     "Although we have presented an O(n log n)
     algorithm for MPT, it remains unknown whether
     the algorithm has an optimum time complexity.
     That will be one of our future research topics."
   

3. Keep sentences simple and short.
   Avoid to write a lengthy sentence that connects clauses one after another.

4. Do not use contractions of two words such as "isn't", "can't", "we've", and "haven't".

5. Do not use the conjunction "so" that is too casual for technical writing.
   Use conjunctions such as "thus", "therefore" and "hence" instead.

6. Organize a paper hierarchically with sections
   that may be decomposed into subsections
   which are further broken down to paragraphs.
   To begin writing a paper, design "Table of Contents" first as an outline of the paper.
   Then, for each section or subsection, try to list important phrases or sentences
   that will turn into paragraphs by elaborating them.

   → Step-wise Refinement in Top-Down Design

   One Paragraph ←→ One Issue (Modularity)

7. "Abstract" summarizes a paper, typically within 100 ~ 200 words.
   An abstract of a paper should be written in order to attract potential readers to its main content.
   It is often used to advertise the paper in research indexes and databases for article search.
   "Keywords and Phrases" are sometimes given next to the abstract.

8. The first section (typically entitled "Introduction") commonly describes some of the following (but not limited to):
   • Overall picture (background) of the topic addressed in the paper
   • Related work with a survey about relevant fields
   • Major achievement / contribution to the relevant fields
   • Organization of the paper at the end of the first section
     
     Example:

         This paper is organized as follows.  Section 2
       presents necessary definitions and notations on
       transactions in a distributed database system.
       Section 3 discusses properties of a transaction
       that are required for fault tolerance.  Based on
       the properties, Section 4 presents a distributed
       algorithm for recovery from failures of
       transactions in a distributed database system.
       Then, Section 5 analyzes the communication
       complexity of the algorithm and compares it with
       those of the previously proposed algorithms.
       Finally, Section 6 summaries our results and
       discusses future directions of our research.
     

9. Sections are serially numbered
   (The first section is "Introduction" and the last section is "Conclusion").
   "Acknowledgments" and "References" following the last section are not regarded as sections and hence their headings are not numbered.

10. Make a paper "self-contained" as much as possible
    in the sense that everything is written in the paper and the paper can be understood by reading it.
    Only exceptions are the knowledge expected to readers (such as knowledge that peers should have) and citations from references listed at the end of the paper.

11. List references in the IEEE Style:
    Refer to pp.5-13 of IEEE Editorial Style Manual
    Each reference should be referred to in the main text at least once by using its label (usually, a serial number).

    Example:

      It is widely known that NP-complete problems
      are time-consuming to solve [2,5].
      .
      .
      .
      References
      [1] E. W. Dijkstra, "A note on two problems
          in connexion with graphs," Numerische
          Mathematik, vol. 1, pp.269-271, 1959.
      [2] M. R. Garey and D. S. Johnson, Computers
          and Intractability:  A Guide to the Theory
          of NP-Completeness, W. H. Freeman and Co.,
          1979.
      .
      .
      .
    

    The IEEE Style requires to list references in the order of occurrences of their citations in the main text,
    although it is also common to list references in the alphabetical order of last names of their first authors
    with tiebreaking by the chronological order of their publication dates.

12. Be careful to use online documents as references.
    They are not always reliable source of information.
    Double check authoritative source of information such as books and journals specialized to the relevant topic.

13. Fully justify text to the both sides.

14. Use proper notations that are common in the presumed field.
    • A set should be written in the set-builder form { x | p(x) }

      while some books use another notation { x : p(x) }.

      A colon is not the common notation in computer science, although it may sometimes be used in Mathematics.
    • Although some books on algorithms use V[G] and E[G] for a graph G, these sets should be denoted by simply V and E, respectively,
      assuming that a graph G is defined as G = (V,E) in advance.

15. A "technical article" unlikely has a table of contents (probably because it is not lengthy) unless it is a thesis or dissertation.
    In contrast, a book usually has a table of contents.

16. A page number starts from the first page of the first section (or chapter in case of a book).
    A page number is an arabic number and printed typically in the center of the bottom margin,
    although the IEEE class and style file IEEEtran.cls layouts a page number at the upper right or upper left in the top margin for an odd or even page number, respectively.
    
    
    If a paper has a cover page, no page number is given to the cover page.
    
    
    If there are pages between the cover page and the first page of p.1, you can put a "roman number" as a page number.

17. "p." stands for a page number and "pp." stands for page-to-page.

18. Place a symbol just next to the noun word defining the semantics of the symbol, e.g.,

      A nonempty set F of n factories fi (1 ≤ i ≤ n)
    

    where F denotes a set and f_i denotes a factory.

19. Pay attention to capitalization of words.
    
    Examples:
    • All reserved words denoting control flow are in lower case.
      
      if ... then ... else ...
      for
      while
    • The data type boolean is commonly in lower case while the technical term "Boolean" should be capitalized in text.

20. Do
    Revise a paper several times.
    • Macro-level revision: Better organization of the paper
    • Micro-level revision: Clearer, simpler sentences/phrases/definitions/notations/etc.

• Writing a Math Phase II Paper by Dr. Steven L. Kleiman (MIT)
• Mathematical Writing by Dr. Donald E. Knuth et al. (Stanford Univ.)

Bad Example

g(x) = ∑∞k=0 ak xk g(x) − 3 x g(x) = ∑∞k=0 ak xk − 3 ∑∞k=1 ak-1 xk = a0 + ∑∞k=1 (ak − 3 ak-1) xk g(x) − 3 x g(x) = a0 = 2 (1 − 3x) g(x) = 2 g(x) = 2/(1 − 3x) = 2 ∑∞k=0 (3k xk)         = ∑∞k=0 2 (3k xk)         = ∑∞k=0 (2⋅3k) xk ak = 2⋅3k f(n) = 2⋅3n

1. Definitions
2. Assumptions
3. Body: A Sequence of Logical Deductions
4. Conclusion

Bad

2y2 = 2x2 + 4x + 2 = 2 (x + 1)2 / 2 = (x + 1)2 = y = x + 1

Good

The following is a given equation consisting of two variables x and y. 2y2 = 2x2 + 4x + 2 By dividing the both sides of the equation by 2, we have the following. y2 = x2 + 2x + 1 The right-hand side of the equation can be factorized as follows. x2 + 2x + 1 = (x + 1)2 Thus, the given equation is transformed as follows. y2 = (x + 1)2 This quadratic equation has two distinct solutions of y in terms of x: y = x + 1 and y = −(x + 1). Note that there are an infinite number of pairs of x and y that satisfy the given equation. Therefore, solutions of the given equation are all pairs of x and y that satisfy at least one of the two linear equations y = x + 1 and y = −(x + 1).

x4 + 1 = 2x2 = x4 - 2x2 + 1 = 0 = (x2 - 1)2 = 0 = x2 - 1 = 0 = x2 = 1 = x = 1, -1

The given equation x4 + 1 = 2x2 is rewritten by moving all terms to the left-hand side as follows. x4 - 2x2 + 1 = 0 By considering x2 as a varible (say, y = x2), we can apply the well-known factorization formula of a quadratic polynomial (say, y2 - 2y + 1 = (y - 1)2) for factoring the polynomial of degree 4 in the left-hand side as follows. (x2 − 1)2 = 0 It implies that x2 − 1 = 0. Then, we get the equation x2 = 1. Thus, the given equation has two roots: x = 1 and x = −1.

1. To express information so that a third party can assimilate it easily and accurately
2. To organize information so that a third party can quickly comprehend it
3. To represent a process of reasoning toward a consequence

• Propositional Logic
• Predicate Logic

• Part 1
• Part 2
• Part 3

• Part 1
• Part 2
• Part 3
• Part 4

How to Compose Sentences Based on Formal Logic

1. Express in formal logic what you have in your mind
2. Translate it into English by referring to typical phrases
3. Revise it by eliminating any ambiguity and simplifying it

Conditional p → q	If p, then q. p only if q. p implies q. When p, q. p only when q. p is a sufficient condition for q. q is a necessary condition for p. q if p. q follows from p. q, provided p. q is a logical consequence of p. q whenever p.
¬ q → p	p unless q.
Biconditional p ↔ q	p if and only if q. p iff q. p is equivalent to q. q is equivalent to p. p is a necessary and sufficient condition for q. q is a necessary and sufficient condition for p.
Universal Quantifier ∀	for all ... all ... for every ... every ... for any ... any ... for arbitrary ... an arbitrary ... for each ... each ...
Existential Quantifier ∃	there exists ... ( such that ... ) there is ... ( such that ... ) there is at least one ... ( such that ... ) there is some ... ( such that ... ) for some ... some ... for at least one ...

Definitions	Let ... be ... Let ... denote ...
Assumptions	Assume that ... Suppose that ... Let ...

Either the election is not decided and the votes have been counted or the votes have not been counted.

For every integer x, if there exists some integer y such that x times y is not equal to x, then x is not equal to 0.

A natural number p is a prime iff p is greater than 1 and there is no factor of p other than 1 and itself p.

1. An integer n is said to be even iff n is a multiple of 2 (i.e., there exists an integer k such that n is equal to 2k).
   Prove that x is even iff x² is even.
   
   
   Hint:
   Let p(x) denote the predicate "x is even" and q(x) denote the predicate "x² is even."
   Then, the above assertion is ∀ x [ p(x) ↔ q(x) ].
   
   
   Prove both ∀ x [ p(x) → q(x) ] and ∀ x [ q(x) → p(x) ].

2. Assume that the universe of discourse is the set ℤ of all integers.
   Prove or disprove the following assertion.
   
   
   "If x and y are odd, then the product of x and y is also odd."

3. Prove or disprove that if an integer is odd, then its square is odd.

4. Assume that the universe of discourse is the set ℤ of all integers.
   Prove or disprove the following assertion.
   
   
   "There is one and only one integer x such that for all y and z, xy is equal to xz."

5. Prove or disprove that the square root of 2 is irrational,
   i.e., the square root of 2 cannot be expressed as a ratio of two integers.

6. Let n be an arbitrary integer.
   Prove or disprove that if n is even, then n³ is even.
   
   
   Remark:
   To prove this rigorously, we should apply predicate logic that is an extension of propositional logic,
   since "n is even" and "n³ is even" are not propositions (they are called predicates).
   In this exercise, you may write a more informal proof.

7. Prove that for every integer x, x is even iff x² is even.

8. Assume that the universe of discourse is the set ℤ of all integers.
   Prove or disprove the following assertion.
   
   
   "If x and y are odd, then the product of x and y is also odd."

9. Assume that the universe of discourse is the set ℤ of all integers.
   Prove or disprove the following assertion.
   
   
   "There is one and only one integer x such that for all y and z, yx is equal to zx."

10. Prove or disprove that if an integer is even, then its square is even.

11. Prove or disprove that the square root of 3 is irrational,
    i.e., the square root of 3 cannot be expressed as a ratio of two integers.

12. Prove the following by induction.
    
    Σⁿ_i=1 i² = n(n + 1)(2n + 1)/6 for all n ∈ ℕ

13. Prove by induction
    Σⁿ_i=1 1/((2i-1)(2i+1)) = n/(2n+1) for every positive integer n.

14. Prove by induction
    Σⁿ_i=1 i(i+1)(i+2) = n(n+1)(n+2)(n+3)/4.

15. Show how you derive the probability that when a fair coin is flipped n times an equal number of heads and tails appear.