poster:how to do,from:
how to do
A definition in mathematics is a precise statement delineating and naming a concept by relating it to previously defined concepts or such undefined concepts as ``number'' or ``set.'' Careful definitions are necessary so that we know exactly what we are talking about. Unfortunately, for many of the concepts in undergraduate mathematics the definition is rather difficult to understand, so often at low levels an intuitive feeling for the meaning of a term is all that is given or required. This intuitive feeling, while necessary, is not sufficient at the college level. This means that you need to grapple with and master the formal statement of definitions and their meanings. How do you do it?
Make sure you understand the definition
This sounds obvious, but it can cause some difficulties, particularly for definitions with complicated logical structure (like the definition of the limit of a function at a point in its domain). Definitions are not a good place to practice your speed reading. In general there are no wasted words or extraneous symbols in established definitions and the easily overlooked small words like
and, or, if ... then, for all, and
there is are your clues to the logical structure of the definition.
First determine what general class of things is being talked about: the definition of a polynomial describes a particular kind of algebraic expression; the definition of a continuous function specifies a kind of function; the definition of a basis for a vector space specifies a kind of set of vectors.
Next decipher the logical structure of the definition. What do you have to do to show that a member of your general class of things satisfies the definition: what do you have to do to show that an expression is a polynomial, or a function is continuous, or a set of vectors is a basis.
Determine the scope of the definition
Most definitions have standard examples that go with them. While these are useful, they may lead you to expect that all examples look like the standard example. To understand a definition you should make up your own examples: find three examples that do satisfy the definition but which are as different as possible from each other; find two examples of items in the general class described by the definition which do not satisfy it. Prove that your five examples do what you think they do---such proofs are usually short, follow the structure of the definition quite closely, and help immensely in understanding the definition. These examples should be neatly written up so that you can refer to them later. Your own examples will have more meaning for you than mine or the book's when it comes time to review.
Memorize the exact wording of the definition.
This step may sound petty, but the use of definitions demands knowledge of exactly what they say. For this reason you can count on being asked for the statement of any definition on an exam. The importance of precise wording should have been made clear by your examples in step 2 and it certainly is essential in the proof of theorems.
Solid knowledge of definitions is more than a third of the battle. Time spent gaining such knowledge is not wasted.
