abstractmath.org
GLOSSARY
Posted 2 January 2009
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
mathematical logic
map
maximize
To maximize a function is to find values of its argument
for which the function has a maximum. Minimize is
used similarly.
Example
The number x = 0 maximizes
the function (discussed here.)
The metaphor behind this usage seems to be: vary the input until you find
the largest value. This is the way most functions are maximized or
minimized (using sophisticated methods of varying the input Newton’s
Method is an unsophisticated example of this). The method you learned in calculus of setting
the derivative to 0 and solving explicitly is rarely of any use in science and
engineering.
mean
To form a definition
"Mean" may be used in forming a definition.
Example
"To say that an integer is even means that it is divisible
by
Implies
To say that an assertion
P means an assertion Q may signify that P implies Q.
Example
"We have proved that
Math
authors do not always make it explicit whether they are using “mean” to give a
definition or to describe an implication.
Average
Of course, mean is also a technical term, referring to the arithmetic average.
minus
The word minus can refer to both the binary
operation on numbers, as in the expression a b, and the unary operation of taking the negative: negating
b gives
b. In current usage in American high
schools, a
b would be pronounced “a minus b”, but
b
would be pronounced "negative b".
The older usage for
b
was "minus b" and many
old fuddy duddy college teachers like me forget and call it “minus b” sometimes.
Don’t
assume a minus sign before an expression makes it negative. The expression “t”
denotes a positive
number if t happens to be
42. Don’t be misled by the fact that we call it
“negative t”. See also subtraction.
“Minus” may also be used to denote set subtraction.
modulo
The phrase “x is the same as y modulo
E” means that x and y are elements of
some set, E is an equivalence relation on the set, and x E y. There are variants to this usage:
¨ x is the same as y mod E.
¨ x = y modulo E ( or mod E).
¨ x is the same as y up to E.
The word arose from a special equivalence relation in number theory, which has two conflicting special
notations in pure math and computing science that I have seen cause confusion
among students.
Mod in number theory
For integers a, b and n, the expression means that a
b is divisible by n. (This relation is an equivalence relation. See the number
theory chapter). For example,
and
. In this usage, the symbol “mod” occurs as
part of a three-place
assertion.
Mod in computer languages
In most computer languages and in computing science texts, the
expression a mod n means the least
nonnegative remainder obtained when a
is divided by n. For example, 23 mod 3 = 2 and
-4
mod 5 = 1. (See number theory chapter
for details.) In this usage, mod is a binary operation on the integers.
Colloquial
mod
People in math-related fields use “mod”
in phrases such as “The administration kept my salary the same modulo [or
mod] inflation”. Presumably the equivalence relation here is something like: “One
dollar in 2002 is equivalent to $1.02 in 2003.”
In particular, a statement such as “The alternating group on three
letters is the same as the cyclic group of order 3 up to
isomorphism” (less often, modulo
isomorphism) refers to the
equivalence relation of two structures being isomorphic.
model
multivalued function
must
You often see "must be" used in math English when
"is" would give the same meaning. It is used with verbs other than
"be" in the same way. This may be to emphasize that the fact being
asserted can be proved from facts known in the context of the discussion.
Other uses of "must" in mathematical discourse are generally
examples of the way the words is used in ordinary discourse.
Example
"If m is a positive
integer and
Example
"Let . If
, then one of A and B must contain two elements of C.”