1
Definitions and Mathematical Terminology
1.1
Welcome
Welcome to the third module in our course on proofs, Definitions and Mathematical Terminology.
Before going on, please do the assigned readings.
1.2
Introduction
If mathematics is thought of as a language, then definitions are the vocabulary and our previous
mathematical knowledge indicates our experience and versatility with the language.
1.3
Definitions
A
definition
in mathematics is an agreement, by all parties concerned, as to the exact meaning
of a particular term.
You have already seen the definition of “
A
⇒
B
” and that definition was
communicated by a truth table. You do not have to accept the definition, but in refusing to accept
a definition you will not be able to communicate with other mathematicians.
Mathematics and the English language both share the use of definitions as extremely practical
abbreviations. Instead of saying “a domesticated carnivorous mammal known scientifically as
Canis
familiaris
” we would say “dog.” Instead of writing down a truth table, we would say “
A
implies
B
.”
However, mathematics differs greatly from English in precision and emotional content. Mathe
matical definitions do not allow ambiguity or sentiment.
Here are some examples of definitions.
•
An integer
n
divides
an integer
m
, and we write
n

m
, if there exists an integer
k
so that
m
=
kn
.
•
An integer
p >
1 is
prime
if the only positive integers that divide
p
are 1 and
p
(note that
this definition makes use of the first definition).
•
A triangle is
isosceles
if two of its sides are equal in length.
•
An integer
n
is
even
if and only if the remainder when
n
is divided by 2 is 0.
Definitions apply not only to the expected mathematical objects, algebraic objects and geometric
objects, but it also applies to statements.
•
Two statements
A
and
B
are
equivalent
if and only if “
A
implies
B
” and “
B
implies
A
.”
•
The statement “
A
AND
B
,” written
A
∧
B
(where “and” is an upward pointing angle), is
true if and only if
A
is true and
B
is true.
•
The statement “
A
OR
B
,” written
A
∨
B
(where “or” is a downward pointing angle) is true
in all cases except where
A
is false and
B
is false.
Note that in all cases the definition stands as an abbreviation for a mathematical expression.
The fact that it is a definition is sometimes indicated by “if and only if,” sometimes by “if,” and
sometimes by “is” or “are.” The context is normally clear.
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 Spring '08
 ANDREWCHILDS
 Math, Logic, Pythagorean Theorem, Natural number, triangle

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