In many math text books you will see *definitions* and theorems.
The definitions
are defining technical words and give a name to a special subclass of
objects. For example, when we study the integers we have a special subset
of positive integers greater than 1 which are called primes. In calculus
we deal mainly with continuous functions or differentiable functions which
are special subsets of functions. Quite often it is these definitions that
lead to proofs of elementary results.
In understanding mathematics, it is important to have a solid
grasp of these definitions. Without that, comprehending a theorem is limited
and attempting to prove a theorem is useless.
Definitions have the form ``*X* is a *blob* if and only if
*condition*''.
This *biconditional* form is used in two ways. First, suppose we have
an object *Y* and want to know if it is a blob. We see if *Y* satisfies the
condition. If it does then *Y* is a blob, otherwise it is not.
Secondly, suppose we have an object *Y* that is a blob. Then we know that
*Y* *does* satisfy the condition and may make use of this fact, in a
proof say.

The majority of theorem statements have two forms. The first is a
conditional statement ``If *condition* then *conclusion*''.
This form is an implication and it means that the conclusion is true whenever
the condition is true. If the condition is denoted by *p* and the conclusion
by *q* then the statement of the theorem is written as .
Consequently, if and only if the logical statement
is a tautology. Often you will see the phrase ``For all''
in statements of theorems.
A statement such as ``For all *x* in *X*, *conclusion*'' is really a
conditional form ``If *x* is an element of *X* then *conclusion*''.
It happens to be true no matter which element *x* we consider.

The second common form of a theorem statement is the biconditional.
This has the form ``*p* if and only if *q*'' or symbollically
. It is equivalent to the two implications
and . Indeed, this is how many of these
theorems are proved (by proving the two implications).
Logically, if and only if
is a tautology.

One important aspect of statements is that they do have equivalent forms. Often, it might be difficult to prove a result as it is stated but it could be easier to prove an equivalent statement. The following table gives a list of common equivalent forms.

We also list some of the English equivalents of the conditional.

Other phrases that appear in theorems are ``There exists'' and ``unique''.
A theorem statement such as ``There exists an *x* in *X* such that
*conclusion*'' is know as an existence theorem. It says that the
conclusion statement is satsfied by something (or the truth set is
non empty). However, the conclusion may be satisfied by * *many
elements of the set. If there is precisely one element of the set that
satisfies the conclusion then the theorem statement would state
``There exists a unique *x* in *X* such that *conclusion*''. Such theorems
are referred to as uniqeness theorems. Our previous statement is an
existence and uniqeness theorem. It is possible to have a theorem which
asserts uniqeness but not existence.

Sun Sep 15 22:27:27 PDT 1996