help with abstract math

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Posted 5 June 2009


Accented characters. 1

Quantifiers. 3

Miscellaneous symbols  4



Accented characters

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 tildeAlso, 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 “?” over a letter is commonly pronounced check by mathematicians.  For example, “ ” is pronouncedz check”.   It is used in Cech cohomology and occasionally in other fields.  The typographical name for this symbol is caron or ek. 


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”. 


¨  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.


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. 


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.

Miscellaneous symbols

  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.


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


¨ 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 .


¨   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.