Math 240          Proof Evaluation #1                Spring,1999
M. Flashman                                                    Due:2-3-99
Let A={ x Î R : x2 + 1 < 10} and B= { x Î R : 0 < x < 2}

Proposition 1: B is a subset of A.

Proof: By the definition of subset, we need to show that if r is an element of B then r is an element of A. Suppose r is an element of B. Then 0 < r < 2, and thus
0 < r2 < 4 so that 0 < r2 +1 < 5. So it should be clear that r is an element of A. EOP.

Proposition 2: A is not a subset of B.

Proof: Consider the number -1, which is a member of A, but is not a member of B. Thus it is not the case that every member of A is a member of B. EOP.

1. Are the statements in propositions 1 and 2 conditional or absolute?
2. List the variables used in these propositions. Indicate what these variables repesent.
3. Are the proofs of these propositions direct or indirect? (see Polya on indirect proof.)
1. If the proof is indirect, state the way in which the argument proceeds.(What is assumed? What is actually demonstrated?)
2. If the proof is direct, does the proof proceed forward or is it mixed with some backward argument? If it has some backward argument, indicate briefly how the original conclusion is altered.
4. Did the proofs explicitly leave some steps for the reader to complete? If so, state what steps the reader is expected to complete. [Optional: complete these steps.]
5. Did the proofs implicitly leave some steps for the reader to complete? If so, state what steps you think the reader is expected to complete. [Optional: complete these steps.]
6. Indicate any parts of the argument that you felt needed greater detail or better connection to the proofs. [Optional: Supply these detail or suggest a better connection.]
7. Overall, do you think these proofs were effective? Discuss briefly the basis for you conclusion.