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.