The Grammar According to West

http://www.math.uiuc.edu/~west/grammar.html

Part IV Terminology and notation #41-53

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41. "Maximal" vs. "maximum" Many mathematicians use these words interchangeably. One can make a useful

distinction by using "maximum" to compare numbers or sizes and "maximal" to compare sets or other objects. Thus a maximal object of type A is an object of type A that is not contained in any other object of type A. A maximum object of type A is a largest object of type A; here "maximum" is an abbreviation for "maximum-sized". For example, in a graph we may speak of "maximal independent sets" and "maximum independent sets"; these are convenient terms for distinct concepts that are both important.

Although this distinction is sensible and has become established in many settings (such as "maximum antichain" and "maximum independent set"), potential confusion can be reduced by using "largest" and "smallest" instead of "maximum" and "minimum". For example, it is harder to misinterpret "a largest matching" than to misinterpret "a maximum matching".

For consistency, then, one should not write "a vertex of maximal degree" or "the maximal number of edges"; that is, "maximal" should not be applied to numerical values. This is consistent with usage in continuous mathematics, where we write that a continuous function "attains its maximum" on a closed and bounded set. 2

41. "Maximal" vs. "maximum" - (1/3)• Many mathematicians use these words interchangeably. • One can make a useful distinction by using "maximum" to

compare numbers or sizes and "maximal" to compare sets or other objects. • Thus a maximal object of type A is an object of type A that

is not contained in any other object of type A. • A maximum object of type A is a largest object of type A;

here "maximum" is an abbreviation for "maximum-sized". • For example, in a graph we may speak of "maximal

independent sets" and "maximum independent sets"; these are convenient terms for distinct concepts that are both important. 3

41. "Maximal" vs. "maximum" - (2/3)• Although this distinction is sensible and has become

established in many settings (such as "maximum antichain" and "maximum independent set"), potential confusion can be reduced by using "largest" and "smallest" instead of "maximum" and "minimum".

(using "maximum" to compare numbers or sizes )• For example, it is harder to misinterpret "a largest

matching" than to misinterpret "a maximum matching".

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41. "Maximal" vs. "maximum" - (3/3)• For consistency, then, one should not write "a vertex of

maximal degree" or "the maximal number of edges"; that is, "maximal" should not be applied to numerical values.

(using "maximal" to compare sets or other objects.)(For example, The degree of diagnosability of the system denotes the

maximal number of faulty processors that can be ensured to identify in the system.)

• This is consistent with usage in continuous mathematics, where we write that a continuous function "attains its maximum" on a closed and bounded set.(using "maximum" to compare numbers or sizes )

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42. Multicharacter operatorsA string of letters in notation denotes the product of individual quantities.

Therefore, any operator whose notation is more than one character should be in a different font, generally roman. This convention is well understood for trigonometric, exponential, and logarithmic functions, and it applies equally well to such operators as dimension (dim), crossing number (cr), choice number (ch), Maximum average degree (Mad), etc.

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42. Multicharacter operators• A string of letters in notation denotes the product of

individual quantities. • Therefore, any operator whose notation is more than one

character should be in a different font, generally roman. • This convention is well understood for trigonometric,

exponential, and logarithmic functions, and it applies equally well to such operators as dimension (dim), crossing number (cr), choice number (ch), Maximum average degree (Mad), etc.

(sin, cos, tan, exp, log)

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43. "Induct on" and "By induction"The phrase "We induct on n" is convenient but not correct. From given

hypotheses, we deduce a conclusion; we don't "deduct" it. When we announce the method of induction, we must instead say "We use induction on n." The verb "to induct" is used when a person is inducted into an honorary society, for example.

A different problem arises in the induction step. When we cite the induction hypothesis, we must write "By the induction hypothesis", not "By induction". To obtain the conclusion for the smaller instance, we are invoking the hypothesis that the claim holds for smaller values; we are not invoking the principle of mathematical induction.

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43. "Induct on" and "By induction” - (1/2)

• The phrase "We induct on n" is convenient but not correct. (For example, “We prove this lemma by induction on n.”)

• From given hypotheses, we deduce a conclusion; we don't "deduct" it. • When we announce the method of induction, we must

instead say "We use induction on n." • The verb "to induct" is used when a person is inducted into

an honorary society, for example.

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43. "Induct on" and "By induction" - (2/2)

• A different problem arises in the induction step. • When we cite the induction hypothesis, we must write "By

the induction hypothesis", not "By induction". (For example, “We now show another bound O(2k) by induction.”)

• To obtain the conclusion for the smaller instance, we are invoking the hypothesis that the claim holds for smaller values; we are not invoking the principle of mathematical induction.

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44. Cliques vs. complete subgraphsThese terms traditionally were used interchangeably in graph theory,

but it is useful to distinguish them. There is a difference between a set of pairwise adjacent vertices in a graph (dual to an independent set of vertices) and a subgraph isomorphic to a complete graph. Both concepts are needed, and the appropriate terms for them are "clique" and "complete subgraph". Thus "clique" should be reserved for a set of vertices, and then the meanings of "clique of size 5" and "5-clique" (the same) are clear. In previous centuries, also "clique" was sometimes used to mean "maximal clique", which should not be done.

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44. Cliques vs. complete subgraphs• These terms traditionally were used interchangeably in graph theory,

but it is useful to distinguish them. • There is a difference between a set of pairwise adjacent vertices in a

graph (dual to an independent set of vertices) and a subgraph isomorphic to a complete graph.

• Both concepts are needed, and the appropriate terms for them are "clique" and "complete subgraph".

• Thus "clique" should be reserved for a set of vertices, and then the meanings of "clique of size 5" and "5-clique" (the same) are clear.

• In previous centuries, also "clique" was sometimes used to mean "maximal clique", which should not be done.

(For example, “For finding cohesion groups, a well-known method is to find out clique which is defined as a set of nodes linked to each other by an edge directly, i.e. a complete subgraph.”) 12

44. Cliques vs. complete subgraphs

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45. Isomorphism classes vs. subgraphsA graph is a pair consisting of a vertex set and an edge set. Paths, cycles, and complete

graphs are graphs whose edge sets are described in specific ways. The notations Pn, Cn, and Kn do not specify a vertex set, and hence in specifying paths cycles, and complete graphs they must refer to the isomorphism classes.

Hence we should never write "a Pn" for a member of that class. We can write that a graph "contains a path with n vertices", because that is a structural description of the subgraph, but we cannot write "contains a Pn" or "consider a Pn in G". We can say "contains ten copies of Pn" to refer to subgraphs that are n-vertex paths; each such subgraph is a member of the isomorphism class denoted by Pn.

Nevertheless, complete strictness about this notation produces very awkward writing. Thus when $H$ is the notation for an isomorphism class, we still write "H G⊆ " to mean that some subgraph of G belongs to the isomorphism class or is "isomorphic to H", even though we are not specifying particular subsets of the vertices and edges of G. graph with n vertices. The reason we accept this slight abuse of the notation "H G⊆ " and not the expression "a Pn" is that "a" is an English word whose meaning and grammatical usage cannot be changed, which emphasizes the difficulty that Pn is not a singular object. 14

45. Isomorphism classes vs. subgraphs - (1/2)

• A graph is a pair consisting of a vertex set and an edge set. • Paths, cycles, and complete graphs are graphs whose edge sets are described

in specific ways. • The notations Pn, Cn, and Kn do not specify a vertex set, and hence in specifying

paths cycles, and complete graphs they must refer to the isomorphism classes. • Hence we should never write "a Pn" for a member of that class. • We can write that a graph "contains a path with n vertices", because that is a

structural description of the subgraph, but we cannot write "contains a Pn" or "consider a Pn in G".

• We can say "contains ten copies of Pn" to refer to subgraphs that are n-vertex paths; each such subgraph is a member of the isomorphism class denoted by Pn. 15

45. Isomorphism classes vs. subgraphs - (2/2)

• Nevertheless, complete strictness about this notation produces very awkward writing. • Thus when $H$ is the notation for an isomorphism class, we still write

"H G⊆ " to mean that some subgraph of G belongs to the isomorphism class or is "isomorphic to H", even though we are not specifying particular subsets of the vertices and edges of G. • graph with n vertices. ???• The reason we accept this slight abuse of the notation "H G⊆ " and not

the expression "a Pn" is that "a" is an English word whose meaning and grammatical usage cannot be changed, which emphasizes the difficulty that Pn is not a singular object.

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46. Proper coloringA k-coloring (or k-edge-coloring) of a graph is a partition of the vertices (or

edges, respectively) into k classes. In combinatorics generally, a k-coloring of a set partitions it into k classes, arbitrarily. This general concept appears in many areas of mathematics, including Ramsey theory, graph decomposition, and chromatic numbers. In the latter context, a proper [edge-]coloring is one in which adjacent [or incident] elements do not receive the same color.

Some authors who write extensively about chromatic number and edge-chromatic number drop the word "proper" and use k-[edge-]coloring for the restricted concept. The minor convenience gained by dropping this word is overwhelmed by the negative influence of introducing inconsistency of terminology in combinatorics. Use "proper k-coloring" when that is what is meant. For other variations, such as "acyclic k-coloring" or "dynamic k-coloring", the adjectives replace "proper" by imposing other restrictions on the k-coloring, so the word "proper" is then no longer needed. 17

46. Proper coloring - (1/2)• A k-coloring (or k-edge-coloring) of a graph is a partition of the

vertices (or edges, respectively) into k classes. • In combinatorics generally, a k-coloring of a set partitions it into

k classes, arbitrarily. • This general concept appears in many areas of mathematics,

including Ramsey theory, graph decomposition, and chromatic numbers. • In the latter context, a proper [edge-]coloring is one in which

adjacent [or incident] elements do not receive the same color.

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Ramsey theory - 世界上任何六個人之中，必定有三個是互相認識或互相不認識的。這樣有趣的事實相信很多人都會聽過，但其實類似的結果還有很多，例如：九個人中一定有三個互相認識或有四個互不認識、十四個人中一定有三個互相認識或有五個互不認識。現在我們嘗試將上述結果加以推廣，探討一下它們背後所蘊含的規律…

46. Proper coloring - (2/2)• Some authors who write extensively about chromatic number

and edge-chromatic number drop the word "proper" and use k-[edge-]coloring for the restricted concept. • The minor convenience gained by dropping this word is

overwhelmed by the negative influence of introducing inconsistency of terminology in combinatorics. • Use "proper k-coloring" when that is what is meant. • For other variations, such as "acyclic k-coloring" or "dynamic k-

coloring", the adjectives replace "proper" by imposing other restrictions on the k-coloring, so the word "proper" is then no longer needed.

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Take a break

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•Log(55)•ｇ•6.9•11002

•Round()•0b•Tan(45)•X2=32+42

(Google “math clock”)

47. Partitions vs. parts

A partition consists of blocks or "parts". Do not use "partition" to refer to the members of a partition. (Students often make this mistake.)

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48. "Pairwise" and "mutually"

Old-fashioned mathematics took the old-fashioned word "mutually" to describe a binary relation satisfied by all pairs in a set, as in "a set of mutually orthogonal Latin squares". In English usage, "mutual" indicates symmetry. Hence modern mathematics should avoid using "mutually" in this way. Instead, the word "pairwise" states exactly what is meant. The change becomes even more important in light of modern terms like "mutual independence" in which "mutual" explicitly does not mean pairwise. (Thus "mutually orthogonal Latin squares" is now ambiguous, but we cannot escape the notation "MOLS(n,k)" in design theory.)

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48. "Pairwise" and "mutually"• Old-fashioned mathematics took the old-fashioned word "mutually" to

describe a binary relation satisfied by all pairs in a set, as in "a set of mutually orthogonal Latin squares".

• In English usage, "mutual" indicates symmetry. • Hence modern mathematics should avoid using "mutually" in this way. • Instead, the word "pairwise" states exactly what is meant. • The change becomes even more important in light of modern terms like

"mutual independence" in which "mutual" explicitly does not mean pairwise.

• (Thus "mutually orthogonal Latin squares" is now ambiguous, but we cannot escape the notation "MOLS(n,k)" in design theory.)

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48. "Pairwise" and "mutually" - MOLS(n,k)

A set of mutually orthogonal Latin squares (1959):• Denition. A pair of n n Latin squares are called orthogonal if when we

superimpose them (i.e. place one on top of the other), each of the possible n2 ordered pairs of symbols occur exactly once.

• A collection of k n n Latin squares is called mutually orthogonal if every pair of Latin squares in our collection is orthogonal.

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48. "Pairwise independence " and "mutually independence "

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49. Disjoint sets

Disjointness is a binary relation. Hence "Consider disjoint sets A1,…Ak" is technically incorrect; we should instead say "pairwise disjoint sets". However, this is a universally understood abuse of terminology, and including the word "pairwise" each time would be ponderous. This principle can be extended to other commonly used binary relations do not make non-binary sense, such as "isomorphic".

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49. Disjoint sets• Disjointness is a binary relation. • Hence "Consider disjoint sets A1,…Ak" is technically incorrect; we

should instead say "pairwise disjoint sets". • However, this is a universally understood abuse of terminology,

and including the word "pairwise" each time would be ponderous. • This principle can be extended to other commonly used binary

relations do not make non-binary sense, such as "isomorphic".

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50. Disjoint union vs. joinIn most of graph theory, it is common to use the notation of

multiplication to denote a graph consisting many disjoint copies of a single component. Thus rK2 is the graph consisting of r disjoint edges. Similarly, Pn1+…+Pnk denotes a linear forest, consisting of k components that are paths with orders n1,…,nk. For consistency, G+H should therefore denote the disjoint union of two graphs G and H. Some authors use G+H to denote the join of G and H, which consists of the disjoint union plus edges joining every vertex of G to every vertex of H. There is other notation available for the join, such as G∨H. However, authors unfamiliar with the join operation (x∨y) in lattices or boolean algebra may not like this. I think an overstruck "+" and "◊" would be reasonable and would suggest the operation, but " " is unavailable ⊕because it often represents symmetric difference. 28

50. Disjoint union vs. join - (1/2)• In most of graph theory, it is common to use the notation

of multiplication to denote a graph consisting many disjoint copies of a single component. • Thus rK2 is the graph consisting of r disjoint edges. • Similarly, Pn1+…+Pnk denotes a linear forest, consisting of

k components that are paths with orders n1,…,nk. • For consistency, G+H should therefore denote the

disjoint union of two graphs G and H.

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G H

50. Disjoint union vs. join - (2/2)• Some authors use G+H to denote the join of G and H,

which consists of the disjoint union plus edges joining every vertex of G to every vertex of H. • There is other notation available for the join, such as G∨H. • However, authors unfamiliar with the join operation (x∨y)

in lattices or boolean algebra may not like this. • I think an overstruck "+" and "◊" would be reasonable and

would suggest the operation, but "⊕" is unavailable because it often represents symmetric difference.

(overstruck "+" and “◊”, , \diamondplus )

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G H

+◊

51. BetweenAn object that is between two other objects separates them; this

is the common mathematical sense of "between". Referring to an edge (or path) with endpoints u and v as an edge "between" u and v is somewhat inconsistent with the rest of mathematics. One can say "an edge joining u and v" instead. In a planar embedding of a graph, an edge shared by the boundaries of two faces is an an edge between the faces.

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51. Between• An object that is between two other objects separates them; this

is the common mathematical sense of "between". • Referring to an edge (or path) with endpoints u and v as an edge

"between" u and v is somewhat inconsistent with the rest of mathematics. • One can say "an edge joining u and v" instead. • In a planar embedding of a graph, an edge shared by the

boundaries of two faces is an edge between the faces.

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52. SetminusThe operator \setminus most often denotes difference of sets.

Hence it is somewhat misleading or old-fashioned (and looks rather pompous) to use it for deletion of elements, as in "G\setminus e". Use "G-e" instead. Also, the notation G\setminus H is easily confused with G/H (especially by students). Of course, there are some contexts (matroids and various algebraic topics), where these notations have special meanings and are quite important, but for simple set difference A-B is preferable.

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52. Setminus• The operator \setminus most often denotes difference of sets. • Hence it is somewhat misleading or old-fashioned (and looks

rather pompous) to use it for deletion of elements, as in "G\setminus e". (“G\e”)• Use "G-e" instead. • Also, the notation G\setminus H is easily confused with G/H

(especially by students). • Of course, there are some contexts (matroids and various

algebraic topics), where these notations have special meanings and are quite important, but for simple set difference A-B is preferable. 34

53. "Left hand side". There is no "hand side", so this expression makes no sense. Even

if one correctly hyphenates to make it "left-hand side", there is still no "hand". Just write "left side".

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The answer of “math clock”

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•Log(55)•ｇ•6.9•11002

•Round()•0b•Tan(45)•X2=32+42

(Google “math clock”)