Math 240          Proof Evaluation #1          M. Flashman

Let A={ x in R : x2 + 1 < 10} and B= { x in 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? [Note: For the purpose of this analysis a conditional statement is a statement that is expressed in the form; "If p then q", or "p implies q". An absolute statement is a statement that is not conditional.]
2. List the variables used in these propositions. Indicate what these variables represent.
3. Does the proof (for Proposition 1 and 2) 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 arguments that you felt needed greater detail or better connection to the proofs. [Optional: Supply these details or suggest a better connection.]
7. Overall, do you think these proofs were effective? Discuss briefly the basis for you conclusion.