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GLOSSARY
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Posted 16 April 2009
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bar
A line drawn over a single symbol is pronounced “bar”. For example, is pronounced “x bar”. It often refers to
the closure (in some sense) of a subset of a set.
Example
“Let X be a topological space with subset Y, and let be the closure of Y in X.”
Other
names for this symbol are macron and
vinculum.
be
The verb “to be” with its inflected forms “is” and “are” has many uses in the English language (see list in Wiktionary). Here I mention a few common usages in mathematical texts.
To assert that an object is an example of a certain type of object
For example, “Every integer is a rational number.” This is essentially the same as “has a property”. In this particular example, you could say, “Every integer is rational”.
Has a property
For example, “The Klein four-group is Abelian.” See property.
To define a property
In
defining a property, the word “is” may connect the definiendum to the name of the property, as
in:
“A group is Abelian if for all elements x and y.”
Note that this is not an assertion that some group is Abelian, as in the previous entry;
instead, it
is saying what it means to be Abelian. It is not always
easy to know whether the author means such a statement as an assertion or as a
definition. In this example the fact
that the word “Abelian” is in boldface
communicates that it is a definition.
See definitions for more about this.
To define a type of object
In
statements such as:
“A
semigroup is a set with an associative
multiplication defined on it.”
the word “is” connects a definiendum with the condition defining it. Note again that it may not be clear whether this is an assertion or a definition. See definitions for other examples.
Is identical to
The
word “is” in the statement
“An
idempotent function has the property that its image is its set of fixed points.”
asserts that two mathematical descriptions (“its image” and “its
set of fixed points”) denote the same mathematical object. This is the same as the meaning of “ =“.
Asserting existence
See existential quantifier for examples.
bracket
This word has several related
usages.
Certain delimiters
In math texts, brackets can mean any of the delimiters parentheses, square brackets, braces, and angle brackets. Some American dictionaries and some mathematicians restrict the meaning to square brackets or angle brackets. The chapter on delimiters has more about brackets in this sense.
Operation
The word
"bracket" is used in various mathematical specialties as the name of an operator (for example, Lie bracket, Toda bracket, Poisson
bracket) on an algebra. The operation called bracket may use square brackets,
braces or angle brackets to denote the operation. The Lie bracket of v and w is
always (as far as I know) denoted by [v,w]. On the other hand, notation for the
Poisson and Toda brackets varies, with different authors using different symbols.
Quantity
The word "bracket" may be used to denote the value of the expression inside a pair of brackets (in the sense of delimiters). For example,
“If the expression is zero, then the
two brackets are opposite in sign.”
I believe this usage may be obsolescent.
but
“and” with contrast
As a conjunction, but typically means the same as “and“, with an indication that what follows is surprising or in contrast to what precedes it. This is a standard usage in English, not peculiar to mathematical English.
Example
“
introduces new property
Mathematical authors may begin a sentence with “but” to indicate
that the
subject under discussion has a relevant property that will now be mentioned. For example, it may be relevant because it
leads to the next step in the reasoning. The property may be one that is easy
to deduce or one that has already been derived or assumed.
This
usage may carry with it no
thought of contrast or surprise. Of course, in this
usage “but” still means “and” as far as the logic goes; it is the connotations that are different.
Example
“We have now shown that m = pq, where p and q are primes. But that implies that m is composite.”
Example
(In a situation where we already know that x = 7):
“... We
have now proved that .
But x is
.”