Math 248 - Spring 2003

This worksheet leads you through the process of studying a theorem. 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 theorem.

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1. Choose a theorem (or proposition or lemma or corollary) 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 theorem number (and name, if it has one) here:

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

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

4. Do you know the precise mathematical meaning of all the words in this theorem? 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:

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

5. Many theorems are in the form "If...then...." If your theorem is in this form, identify the hypotheses and the conclusion and write them below:

 

 
 
 

Hypotheses:
 
 
 
 
 

Conclusion:
 
 
 
 

6. If the theorem is not in the form "If...then...", try to categorize its form. Some possibilities are "There exists?" or "The following are equivalent?" or "Object A = Object B". There are other possibilities as well.

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

7. Draw a picture or diagram for this theorem if applicahle.

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

8. Think of some examples that satisfy the hypotheses of the theorem. Write them below. If possible, verify that they satisfy the conclusion of the theorem. If your theorem is of a different form, think of examples as appropriate.

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

9. Think of an object which does not satisfy the conclusion of the theorem. Can you verify that it also does not satisfy the hypotheses of the theorem? Adapt as necessary for theorems of other forms.

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

10. Write the converse of the theorem and try to determine whether or not it is true. If it is not true, try to come up with a counterexample.

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

11. Before reading the proof of the theorem, think of how you would try to prove it. How would you start? Can you perhaps prove a special case of the theorem? (Even if you don't come anywhere near proof, this exercise can greatly increase your understanding of the theorem and its proof).

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

12. Try to come up with a generalization of the theorem and write it below. Try to come up with some evidence (usually examples) as to whether or not the generalization is true.