Posted 3 September 2007
OTHER SYMBOLS incomplete
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 bar, check, 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.
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.