The Grammar According to West http://www.math.uiuc.edu/~west/grammar.html

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The Grammar According to Westhttp//www.math.uiu
c.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.

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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.

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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 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.

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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 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.

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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.)

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44. Cliques vs. complete subgraphs
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45. Isomorphism classes vs. subgraphs

• 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.
• 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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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.

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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 coloring

• 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.
  
• 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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46. Proper coloring - (1/2)
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• 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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46. Proper coloring - (2/2)
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• 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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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. join

• 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, Pn1Pnk denotes a
  linear forest, consisting of k components that
  are paths with orders n1,,nk. For consistency,
  GH should therefore denote the disjoint union of
  two graphs G and H. Some authors use GH 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.

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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, Pn1Pnk denotes a linear forest,
  consisting of k components that are paths with
  orders n1,,nk.
• For consistency, GH should therefore denote the
  disjoint union of two graphs G and H.

?
r
G H
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50. Disjoint union vs. join - (2/2)

• Some authors use GH 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 )

G H
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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
  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. 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". 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.

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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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