Definitions:

Definitions in mathematics are, formally, Aif and only if@ statements with appropriate quantifiers. Thus they establish the logical equivalence of two statements B one the statement being defined, and the other the definition.

Examples:

The following is a definition taken from my old Number Theory Text ( 8 1972, and yes, that does make me feel old):

Let a and b be integers, a0. The statement Aa divides b@ means that there exists an integer x such that ax = b. . . .The notation a|b is used to mean Aa divides b.@

This becomes formalized as:

ab (a 0  (a | b if and only if x (ax = b) ))

Notice that this is a proposition. Since it is given as a definition, we are expected to take it as a true statement. Notice also the various parts of the definition:

ab / (a 0 /  / (a | b / if and only if / x (ax = b)))
Establishes the truth for all integers, Except that we want to make one exception. So. . . . / This is a way of limiting the definition to just numbers that make sense for it, i.e., just to nonzero a=s / This is the two-variable predicate that is being defined / Establishes that the predicate a|b on the left is exactly logically equivalent to the one on the right. / This is also a predicate in two variables, that defines the one on the left.

Notice the translation

ab / (a 0  / (a | b / if and only if / x / (ax = b) ))
Let a and b be integers,
a0.
The statement Aa divides b@
means
that there exists an integer x
such that ax = b.

Having defined a basic concept a|b we can now go on to define a whole lot of wonderful things:

Example:

AThose integers a such that a|b are called divisors of b.@

Formalized:

ab (a is a divisor of b if and only if a|b)

Example:

A positive integer n is called prime if n >1 and its only divisors are 1 and n.

Formalized:

n (n > 0  (n is prime if and only if (n > 1 and x ((x is a divisor of n)  x = 1 or x = n)) ))

(Whew!)

Notice the translation:

n / (n > 0  / n is prime / if and only if / (n > 1 and / x ((x is a divisor of n)  x = 1 or x = n)) ))
A
positive
integer n
is called prime
if
n >1 and
its only divisors are 1 and n.

Let=s try another definition or two:

AWe say that a group G is cyclic if there exists an a  G such that every x  G is a power of a.@

Formally,

G ( G is cyclic if and only if (a (a  G and x (x  G n (n  Z+ and x = an)))))

Translation:

G / G is cyclic / if and only if / a / (a  G and / x (x  G  / n (n  Z+ and x = an)))))
A group G
is cyclic
if
there exists an a
a  G such that
every x  G
is a power of a.

Example:

A real number c is a least upper bound for a set S if x  c for all x in S, and provided c  b whenever x  b for all x in S.

Formally,

cS ( c is a least upper bound for S if and only if (x (xS x  c ) and (x (xS x  b)  (c b))))

Herewith the translation:

cS / c is a least upper bound for S / if and only if / ((x(xS  / x  c) / and / (x (xS  / x  b) /  / c  b )))
A real number
c is a l.u.b for a set S
if
x  c
for all x in S,
and provided
c  b
whenever
x  b
for all x in S.