5.9 -- Mathematical Writing

What to call a result

Lemma: stepping stone toward proving a theorem
Sometimes a lemma achieves fame of its own: Pumping Lemma

Corollary: a direct or easy consequence of a lemma or theorem; a corollary states a result more specific than a lemma or theorem more specific, in contrast to a generalization

Conjecture: an assertion that the author thinks may be true but has been unable to prove or disprove

Hypothesis: In mathematics, a statement regarded as true in subsequent deductions; in statistics, a statement to be accepted or rejected based on a statistical test

Math: Inductive hypothesis

Statistics: Null hypothesis

When should a result be given one of the above labels (i.e., theorem, lemma, or corollary)?

• Some authors develop ideas as a stream of theorem/proof pairs.

• Other authors use a less formal exposition style, saving key results for formal treatment.

Writing a Proof

• A proof can convince the reader of the truth of an assertion.

  In this case, the proof can be more structured and must be complete.

• A proof can convey an interesting property about the system for which one is conducting the proof.

  In this case, the proof must convey the key ideas and insights, possibly organized in a less structured form or written as a proof outline.

Miscellaneous comments:

• Help the reader keep track of where they are in the proof.

• Clearly state the inductive hypothesis, if used.

• Clearly identify whether you are proving the if or the else part, for iff theorems.

• Omit unnecessary or tedious details, or delegate them a lemma

• Mark the end of the proof if proofs run for multiple paragraphs. Use a halmos (i.e., a box) or use Q.E.D. (Latin: quod erat demonstradm = which was to be demonstrated).

Examples...

The aim/idea is to...

The trick of the proof is to find ...

... is the key relation

To obtain ... a little manipulation is needed.

It is easy/simple/straightforward to show that ...

Some tedious manipulation yields ...

First, we establish that ...

Our problem reduces to ... Finally, we have to show that ...

Examples of proof styles...

Chandy and Misra, Parallel Program Design: A Foundation , Addison Wesley, 1988.

L. Lamport, How to Write a Proof, DEC SRC technical report, Feb. 1993.

L. Lamport, How to Write a Long Formula, DEC SRC technical report, Dec. 1993.

The Role of Examples

Discuss examples before the general case:

"Suppose you want to teach the "cat" concept to a very young child. Do you explain that a cat is a relatively small, primarily carnivorous mammal with retractable claws ... ?

I'll bet not. You probably show the kid a lot of different cats, saying "kitty" each time, until it gets the idea.

To put it more generally, generalizations are best made by abstraction from experience." -- R. P. Boas

Select your examples carefully! There is a trade-off between examples that are

• simple enough for the reader to entirely understand, and

• rich enough to illustrate multiple concepts of interest.

One method of exposition is to weave an example with general properties in the form of theorems/proofs.

Definitions

Why define a term? If you do not use the term more than two times, consider omitting the term.

Where?

Put the definition before use.

Try to put the definition immediately before use. (Do not put them 10 pages back!)

Use redundancy for terms that have not been used recently:

Page 1: Let C be the capacity of a link in bits/sec.

Page 9: Throughput is limited by link capacity C.

Avoid long blocks of definitions.

How?

There may be more than one definition -- choose one that is understandable, but also concise, precise, and consistent with other definitions.

"If" is taken as "if and only if" in a definition, so do not write "if and only if".

A graph is connected if there is a path from every node to every other node.

Use italics to clearly identify that the sentence is defining a term.

Notation

Minimize your use of notation:

If you do not use a symbol more than two times, eliminate the symbol

Strive for simple, intuitive notation.

Do not use similar-looking symbols that the reader might confuse: X with a zero versus the letter O as a subscript.

Use notation conventions. Conventions that cross fields are:

• epsilon , delta for small quantities

• i,j,k,m,n for integers

• pi, e for fundamental constants

Use different classes of symbols for different types of objects.

Example: Upper case symbols denote matrices.

See Table 3.1 in [H] for the history of common symbols.

Words versus Symbols

• Use symbols if the idea is too cumbersome for words.

• Use a symbol for a concept about which you wish to make a precise mathematical statement.

• Use words if the reader will have trouble grasping the meaning or essential feature of an expression.

• Use words if they are shorter than symbols.

Placement of Symbols

Separate math symbols by punctuation marks or words:

Bad: If x>1 f(x)>0.

Fair: If x>1, f(x)>0.

Good: If x>1 then f(x)>0.

"The" or "A"

Only use "the" when it refers to a unique object. In contrast, "a", the indefinite article, refers to one of a set of objects.

In mathematics, misuse of "the" can lead to a false implication.

Example:

Bad:
Under what conditions does the iteration converge to the solution of f(x)=0?

Good:
Under what conditions does the iteration converge to a solution of f(x)=0?

Miscellaneous Points

1. Set symbols in italics font.

2. Do not set standard mathematical function names in italics:
   sin, cos, max, gcd, lim, ...

   Example:

   tanx is the product of four scalars
   tan x is the tangent of x

3. In TeX, do not write $this$ to put a word in italics, because it uses spacing appropriate to the product of symbols.