Understanding Mathematical Definitions

Math 248 - Spring 2003

Understanding mathematical definitions is crucial for understanding and writing mathematical proofs. When reading and doing mathematics, you should find yourself referring back to the definitions very often. When you come to a definition in your reading, stop and try to understand it thoroughly before continuing.

This worksheet leads you through the process of studying a definition. Do the first four steps in order; the others can be done in any order. You should go through a similar process whenever you encounter a new definition. After doing this work, you will know the definition so well that you will probably not need to "memorize" it.

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  1. Choose a definition from the first four chapters of the textbook, preferably one which seems a bit difficult to you. (Chapter 5 is okay if you wish). Write the page number and the name of the object being defined here:

    2.  
  2. Read the definition several times.
  3. Write the definition here, exactly as if appears in the book:

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  4. Do you know the precise mathematical meaning of all the words in this definition? If there's any you need to review, look them up, using the index of the book. Write down the definitions of these words here:

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  5. Draw a picture or diagram for this definition if applicahle.

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  6. Think of some examples of the object being defined. Write them here:

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  7. Think of some examples of objects which come close to satisfying the definition, but which do not satisfy the definition. For example, if you are studying the definition of "function continuous at a point," think of a function which is defined but not continuous at a point. Write these down here:

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  8. If the notation does not appeal to you, rewrite the definition using notation that you prefer. Be sure you don't change the meaning of the definition!

    9.  

     
     
     
     
     
     
     
     
     
     
     
     
     
     

  9. If you know another definition for this object, or if you already have an intuitive understanding of the concept, try to verify whether or not the definition agrees with your previous understanding. For example, if you are studying the definition of the limit of a function at a point, you may already know this means something like "as x gets closer and closer to the point in question, f(x) gets closer and closer to the value of the limit." Be careful - it often happens that one's previous understanding is not quite right! Write your work on this below: