abstractmath.org

help with abstract math

Produced by Charles Wells.  Home.   Website Contents     Website Index   
Back to Languages Head

Posted 3 September 2007

OTHER SYMBOLS  incomplete

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 are barcheck, hat and tilde.

Example

Let X be a subset of a space S.  Then define  to be the closure of X in S.  For closures, bar is the most common, but we could also use  or .

See also prime.

Quantifiers

 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.  It is not common to see it in printed texts except in logic courses.

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.”

Miscellaneous symbols

 

  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 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.) 

 

 

 

TO BE WRITTEN

 

   Various types of inclusion.

¨     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.

¨      has two meanings in math writing and authors may not tell you which one they are using:

a)   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.

b)   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.

 

¨      can mean three things:

c)    

d)    but .

e)   When A and B are statements (not sets)  may mean “If B, then A”.

¨      means “  but  ”, which is one of the meanings of “  ”The advantage of the symbol “  ” is that it is unambiguous.

¨      means the same as .

¨      is the negation of  in whichever sense the author is using it.

¨      is the same as , whatever it means.