Posted 5 June 2009
OTHER
SYMBOLS
Mathematicians frequently use an accent
to create a new variable from an old one, usually to denote a mathematical object with some specific functional
relationship with the old one. In math, the most commonly used accents over the letter are arrow, bar, check,
hat and tilde. Also, prime and
star may be used after the letter as a superscript, and slash may be used across a relational symbol.
In the list
below, I mention some very common meanings, but mathematicians may and do define
other meanings for the symbols.
The expression 
usually denotes a vector. It may be pronounced “v arrow”. A variant is . In print, vectors are very often
printed in boldface, for example v, but the arrow version
is useful at the blackboard. The arrow
is also used stand-alone in arrow
notation.
For a variable x, is pronounced “x bar”. In math, the bar is commonly used to denote
the mean of a random variable or the closure (however it is defined) of a
substructure of a mathematical structure. The typographical name of the symbol is macron.
The symbol “ˇ” over
a letter is commonly pronounced check
by mathematicians. For example, “ ” is pronounced “z check”. It is used in Čech cohomology and occasionally in other
fields. The typographical name
for this symbol is caron
or háek.
The expression is pronounced “x hat”. It is used in many ways in math, including
for closures (like bar)
and for the sample mean of a random variable.
The typographical name of the symbol is circumflex.
The symbol " ~ " is pronounced "tilde" (till-day or
till-duh), or informally "twiddle" or "squiggle".
¨
It may be used over a letter to
create a new variable., for example " ".
This would be pronounced "A tilde", "A twiddle" or "A squiggle".
¨
are all used to denote a binary relation. More about this below.
¨
Before a number the tilde is used
to mean “approximately”. “~42” means
“approximately 42”.
Other uses are discussed here.
The symbol “ ” is pronounced “prime” or “dash”. For example, is pronounced “x prime” or “x dash”. is pronounced “x double prime”. The “dash” pronunciation is used mostly
outside the United States.
¨
If x is a variable, ,
and so on may be defined as new variables of
the same type.
¨
If f is a function on the reals or the complex numbers, usually denotes its derivative.
The example of a detailed proof here
shows the prime and double prime in use.
See more about the prime symbol here.
The asterisk “*” may be used independently or as a modifier
of a variable. In either case (except
when it denotes multiplication) it is pronounced “star”.
Modifiers
¨
If a is a variable a*
may denote the result of applying a specific involution to a, for example the complex conjugate or
the adjoint.
¨
If a is a character, a* may denote any number of repetitions
of the character.
Used independently
¨
A “*-algebra” is an algebra with a
designated involution called “*”.
¨
In programming languages “*” is
used to denote multiplication. More here.
¨
The word “star” may be used as a mathematical adjective without being
associated with asterisks, as for example with star polyhedra.
A slash across a
relational symbol means the resulting statement is false. For example, means that the statement x = y is false; in other words, x
is not equal to y.
is the universal
quantifier. The meaning is discussed
in detail here.
is the existential
quantifier. For meaning, see here.
Usage
Many mathematicians use the symbols and at the blackboard as abbreviations. Except in books and papers on logic, it is
not common to see them in printed texts.
Pronunciation
Pronouncing expressions using the two quantifiers involves a
difference in syntax.
¨
is pronounced “For
all n, n is even.”
¨
is pronounced “There
is an n
such that n is even.”
See also the articles on notation for connectives
and negation.
This can mean multiplication of numbers (more here), multiplication of matrices, the cartesian
product of mathematical objects, or the vector product of 3-dimensional vectors.
The
phrase “ ” means x
is an element of the set S. Discussed here.
Usage
Some mathematicians write this symbol as (the Greek letter epsilon)
and even sometimes call it “epsilon” but it is really intended to be a
different symbol.
The empty set.
Usage
¨
The symbol " " is not the Greek letter phi, written . (You will occasionally hear mathematicians
refer to the empty set as “phi”, which is historically incorrect).
The symbol is not an example of
the use of slash.
|
¨ The symbol " " is not the number 0, even though it
looks like the form of the number zero sometimes written by older printers. Some approaches to foundations define the number 0 to be the empty set, but that does not mean you should think of them as the same thing. See Literalism.
These symbols all denote binary relations, in
most cases relations that are reflexive and symmetric..
The equals sign “=” has a
standard meaning in math: “x = y” means x and y are two different
names for the same mathematical object. However, in some programming languages it is
an assignment operator,
and the intent of an expression containing an equals sign may
vary. See equation.
The other signs may indicate various forms of approximately equal, or may
denote an equivalence relation. In
particular, see congruence and equivalence.
This means “is defined to be” and the symbol
is called colon
equals. For example, means that f(x)
is defined to be and means that A
is defined to be the set {1,3,5,6}.
The scope of definitions given this way are usually small, for example
the current paragraph or section.
This usage originated in computing science but some
mathematicians now use it, especially at the blackboard.
Various types of inclusion.
The two meanings of “ ”
“ ” has two meanings in math writing and, at
least in research literature, authors are not likely tell you which one they are using.
¨
It may mean exactly the same thing
as “ ”, including
the possibility that A = B. This meaning is more common in research
papers that in textbooks but it occurs in textbooks too.
¨
It may mean but ,
in analogy with “ ” and “<”.
This meaning is common in textbooks and seems to be universal in high
school math texts.
When you see “ ” make sure you know what the author means.
The three meanings of “ ”
“ ” can mean three things:
¨
¨
but .
¨
When A and B are statements
(not sets) may mean “If B, then A”. More about this here.
The others
¨
: A
is included in B, or A is a subset of B. Includes
the possibility that A = B.
¨
: B includes
A, B is a superset of A, or any of the phrases meaning . Again, A
can equal B.
¨
A B means “ but ”, which is one of the meanings of “ ”. The advantage of the symbol “ ” is that it is unambiguous. It is not very widely used, but it ought to
be.
¨
means the same as .
Negations
¨
means there is an element of A that is not in B.
¨
means every element of A is in B and there is an element of B that
is not in A.
