abstractmath.org GLOSSARY
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say
sign
The word sign is used to
refer to the symbols “ +” (the plus sign) and “ ”
(the minus sign).
The word is also used to refer to the question of whether an expression
represents a real number that is positive or negative. For example, if
,
you may say
“For negative x, f(x) and f'(x) are opposite in sign.”
Observe
that for negative x, the expression f'(x) denotes a
negative number even though the expression has no negative sign in it.
The word sign is also
sometimes used to refer to other symbols, for example “the integral sign”.
subscript
This should be moved to the symbols chapter.
An expression such as commonly means that a is the entry indexed by 2 in a sequence. Since a sequence is a function on its
index set, authors may refer to
as a “function on I”, and use language such as “a
is increasing in I”.
subtract
In ordinary English, if you subtract from a collection you make
it smaller, and if you add to a collection you make it bigger. In math, “adding”
may also refer to applying
the operation of addition; but then a +
b is smaller than a if
b is negative,
and subtracting b from a makes the result bigger if b is negative.
Both
these usages (adding to a collection and applying the operation of addition)
occur in mathematical writing. See minus.
such that
For an assertion
P, a phrase of the form “ c such
that P(c)” means that P(c) holds.
Examples
¨
“Let n be an integer such that .”
means that in the following assertions that refer to n, one can assume that
.
¨ “The set of integers n such that “ refers to the set
. (See setbuilder
notation. and the.)
Remarks
¨
Note that in
pronouncing the phrase “such that” is usually inserted.
This is not done for the universal
quantifier. So “
x(x>0)”
is pronounced “There is an x such
that x is greater than 0”, but “
x(x>
¨ Yes, I know that “ x(x>
sufficient
P is sufficient for Q
if the statement “If P, then Q” is true. You
can also say P suffices
for Q. The idea behind the word is
that to know that Q is true it is
enough to know that P is true. See
conditional
assertion.
superscript
This should be moved to the symbols chapter.
An expression such as may denote a
to the nth power, but it may
also indicate that it is the nth
entry in a sequence indexed by superscript i. Typically superscript indices (see plural)
occur in conjunction with subscript indices, but it is perfectly possible and
common in some fields for superscript indices to occur without subscripted
ones.
suppose To be written
symbol manipulation
Symbol manipulation is the use of algebraic rules, or rules of
some other computational system, to change a symbolic
expression to an equivalent one.
For example, you can change to the equivalent expression
using the distributive law for algebra, and
you can change
into
using an integration rule.
symbolic logic
Symbolic logic is the study of statements and proof as a mathematical system. It is also called formal logic or mathematical logic. (Some authors give these phrases slightly different shades of meaning). Assertions, connectives, quantifiers and rules of deduction are typically represented using symbols, and the symbolic system developed this way is studied as a mathematical object.
Examples
Assertions containing a variable x might be represented as P(x) and Q(x). Then expressions built up out of these may be represented in one of these ways (this list is not exhaustive):
means “for all x, P(x)” (universal quantifier).
means “there is an x for which P(x) is true”
(existential quantifier).
means “P(x)
or Q(x)” (also written P(x) + Q(x)).
means “P(x)
and Q(x)” (also
written P(x)Q(x) or P(x) & Q(x)).
means “If P(x),
then Q(x)” (also written
or
).
References
These websites provide more complete descriptions of this symbolism and of the theory behind it.
¨ Suber
¨ Williams
.