The Grammar According to West math.uiuc/~west/grammar.html

1. The Grammar According to Westhttp://www.math.uiuc.edu/~west/grammar.html Mathematical style #10 Terminology and notation #28-40

2. 10. Mixing words and notation • Words cannot be compared with notation via a relational symbol. • "Consider a graph G with maximum degree ≤ k". (X) • Grammatically, the sentence does not indicate where the inequality starts. • If one side is written in words, then the relation must also be written in words. • This restriction is a logical consequence of treating the notation as a unit; in that sense, the sentence above says that the maximum degree of G equals the expression "≤ k". • "Consider a graph G with maximum degree at most k". (O) • "Consider a graph G with maximum degree (G) where(G) ≤k". (O)

3. 10. Mixing words and notation • The same principle applies to logical symbols. In written mathematics, do not use the symbols (∃,∀,⇒,iff) to substitute for words in sentences. • A graph is bipartite iff its chromatic number is less than or equal to 2. • A graph is bipartite if and only if its chromatic number is less than or equal to 2. (O) • Shorthand notation used to save space on lecture slides need not follow these restrictions, since the slides summarize the lecture and are accompanied orally by sentences.

4. Terminology and notation (especially in discrete mathematics) 28-53

5. 28. Definition symbol ":=" • Some mathematicians use this symbol to indicate that the preceding symbol is being defined to mean the subsequent object. • "Let [n]:={1,…,n}.” • Reading ":=" requires thinking "be defined to be" when preceded by "let“. • "Consider a coloring of [n]:={1,…,n}.” (X) • Double-Duty Definition. • This awkward notation is never needed and encourages grammatical errors. • "Consider a coloring of [n], where [n]={1,…,n}.“ (O)

6. 29. "Such that" in set definitions: ":" vs. "|" • "{3n+1: n∈N}". (O) • As in English, there should be space after the colon but not before it. • For many reasons, the colon “:” is a far better choice than the vertical bar “|” to mean “such that” in a “notation/condition” definition for a set. The vertical bar is heavily used in mathematics, most notably for size of sets, but also for divisibility and other purposes. • "{|A|||A|||B|}”. (X) • purports (appear or claim) to describe the set of sizes of sets A that divide the sizes of sets B. • Since "such that" is not a binary operator, this usage should be expressed in TeX using "\colon\," instead of ":".

7. 30. Sequences(數列), series, and lists • In mathematics, a sequence is a function whose domain is the set of natural numbers (perhaps with a shift of the initial element). • The Fibonacci Sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... • An n-tuple is a list of length n. • Discrete mathematicians abuse this term in using it for an ordered finite set. • "a sequence of length n“. (X) • "degree sequence" should be changed to "degree list“. (To avoid this problem, one can sometimes refer to the "vertex degrees”)

8. 30. Sequences, series(級數), and lists • In English a "series" usually consists of finitely many occurrences in order, as in the "World Series" or the title "A Series of Unfortunate Events". • In mathematics a series is an infinite sum. • Geometric Series • Taylor Series

9. 31. Listing the second element of a list • The expression "v1,v2,…,vn" for an indexed n-tuple is a style used to suggest that the elements are indexed by the first n positive integers with no skips. • However, the most natural interpretation of the expression "v1,…,vn" is exactly the same. • The appropriate convention is that indices in a list are consecutive unless explicitly indicated otherwise. • Another reason to eliminate v2 from the expression is that "v1,v2,…,vn" forbids the possibility n=1.

10. 32. A list with relations • The sentence "Let x1≤…≤xn be a list of integers" is a Double-Duty Definition; the writer attempts simultaneously to introduce notation for the elements of a list and to impose inequalities on them. • The expression "x1≤…≤xn" denotes a set of relations, not a list; what is meant is "Let x1,…,xn be integers such that x1≤…≤xn." • To avoid repeating the notation, it is better to write "Let x1,…,xn be integers, indexed in nondecreasing order." • Similarly, a chain of sets under inclusion is a list A1,…,Ak such that A1⊆…⊆Ak; the expression "A1⊆…⊆Ak" is not itself a chain.

11. 32. A list with relations • The ellipsis in an indexed list with relations should be vertically centered on the line ("\cdots" in tex). • x1≤…≤xn • The ellipsis in an indexed list separated by commas should be on the baseline ("\ldots" in tex). • "v1,v2,…,vn"

12. 33. Variable equal to list • Many mathematicians write "for m=1,2,…,n" (with or without the "2") to mean "for m∈{1,…,n}" or "for 1≤ m≤ n". • The expression "for m=1,…,n" is mathematically incorrect; it sets the value of m to be a list of numbers. • The same principle applies to writing "i=1,2" to name two cases; this should be i∈{1,2}.

13. 34. Big Oh • The expression "f(n)=O(n²)" does not mean that the value f(n) equals the set represented by the notation O(n²). • What is meant is "f(n)∈ O(n²)"; Knuth has written at length on this subject. • An alternative that is roughly correct is to be more informal, writing "f(n) is O(n²)", in which "is O(" can be read as "is on the order of". • Since it is convenient to do arithmetic with these classes of functions, this problem will not go away. An unsatisfying compromise is to use the membership symbol where the grammar of computation permits, in order to ensure that the meaning of the concept is understood.

14. 35. Operators vs. constants • We never use f to denote the value of a function f at a point x (f(x)). • The same principle applies to graph parameters and other operators. For example, the maximum degree of a graph G is denoted Δ(G). • Here Δ is a function, not a number, and hence Δ should not be used to denote the value of the function Δ on a particular graph.

15. 35. Operators vs. constants • It is tempting for mnemonic reasons to write "We write V=V(G) and Δ=Δ(G)". Admittedly, this usage is not confusing when discussing only one graph at a time; the difference between a graph invariant and a real-valued function is that we rarely focus on the value of a real-valued function at just one point. Nevertheless, it is rare that a paper discusses only one graph, and hence it is better to use V(G) and Δ(G)for objects associated with G.

16. 35. Operators vs. constants • The problem is particularly bad with Δ, since this character also occurs in mathematics as a difference operator. • One often sees "Δn" meaning the change in the value of n, so one should not use "Δn" to mean the maximum degree times the number of vertices in a graph. • In my textbook I violated this principle by using n(G) and e(G) for the numbers of vertices and edges in a graph G while using n for the number of vertices of a particular graph and e as a particular edge; the error will be corrected in the third edition.

17. 36. Hyphenation • A k-edge connected graph is a connected graph with k edges (compare with "n-vertex connected graph"); the meaning is different from "k-edge-connected graph". • When the hyphen is missing, "k-edge" modifies "connected graph" because adjectives modify only nouns, not other adjectives. (Similarly, a non-specialist reader would think that a k-edge coloring is a coloring of k-edges, not a coloring of edges using k colors.)

18. 36. Hyphenation • Two-word terms used as single concepts to modify nouns must be hyphenated when so located (without the hyphen in this sentence, we would be discussing two "word terms"). • Double-Duty Definition • "A well-known theorem is a theorem that is well known."(O) • "k connected graphs" would be k graphs that are connected, in contrast to "k-connected graphs". • Adverbs behave differently, since they can modify adjectives; for example, we may write "upper chromatic number" without hyphens.

19. 36. Hyphenation • In this setting, we do not need "vertex" as an adjective to specify "connectivity" or "chromatic number", but we add "edge" for the analogous edge concept. We then hyphenate "edge-connectivity" and "edge-chromatic number". • This makes sense because in both cases the problem for edges is a special case (for line graphs) of the general coloring or connectivity problem. • When comparing "edge-coloring" and "list coloring", the difference is then that we are not coloring the lists, so the format of the term is different from that for edge-coloring.

20. 36. Hyphenation • When an expression involving addition or subtraction is used as a parameter modifying a noun, it should be enclosed in parentheses. • For example, write "(k+1)-connected graph", not "k+1-connected graph".

21. 37. a-b path • In "a-b path", "a-b" is not a word and has no notational meaning by itself. • Even worse, often a-b is treated as a mathematical expression in tex and is typeset using a long minus sign with extra space around it. • $a-b$ : ab • The intent is to specify a path with endpoints a and b. Thus a and b are parameters designating a certain type of path. • Under the principles of hyphenation above, there must be a hyphen between "b" and "path". • Hence the correct notation is "a,b-path".

22. 38. Graphs are not sets • When h is a vertex in a graph G, it makes no sense to write h∈ G, since h could just as easily be an edge. A graph consists of a vertex set and an edge set; one should write v∈ V(G)ande∈ E(G). This is also the reason why the convenient notations |G| and ||G||are mathematically inconsistent for the order and size of a graph. • The term "order" for the number of vertices of a graph is not as popular as it once was. Some readers find it confusing and prefer "number of vertices". On the other hand, it is very convenient, while overuse of "number of vertices" becomes quite awkward.

23. 39. Directed graphs and hypergraphs • In a digraph, the edge set consists of ordered pairs. • E(G)={(a,b),(b,c)} • The redundancy of saying "directed edge" or "directed path" or "directed cycle" is not helpful, as it suggests that the digraph contains such objects that are not directed (the term "weak path" is available for a path in the underlying undirected graph). • Using these terms also eliminates the possibility of making statements that hold for both graphs and digraphs, like Menger's Theorem.

24. 39. Directed graphs and hypergraphs • Similarly, one should not use "hyperedges" to refer to the edges of a hypergraph. • Hypergraphs generalize graphs by allowing edges to have arbitrary size. • Calling them "hyperedges" eliminates the possibility of saying that graphs arise as a special case, since graphs have edges, not hyperedges.

25. 40. Connected components • Unnecessary redundancy has similar disadvantages. We should not speak of the "connected components" of a graph, because there are no disconnected components of a graph. • Writing "connected components" suggests that there are components that are not connected.