Definitions: (1) For a,b real numbers with a < b, (a,b) ={ x : a < x < b}

(2) A set of real numbers, O, is called an open set if and only if for any number x that is a member of O there are  numbers a and b with a < b so that x is a member of (a,b) and (a,b) is a subset of O.

Proposition 1: For any real numbers r and s with r<s, (r,s) is an open set.

Proof: Suppose x is a member of (r,s). Then using a=r and b=s in the definition of an open set, there are numbers a and b with a < b so that x in (a,b) and (a,b) is a subset of (r,s). EOP.

Proposition 2: If U and V are open sets and Z is the  the union of U  and  V, then Z is an open set.

Proof: Suppose x is in Z. Then either x is in U or x is in V.

Suppose x is in U. Then since U is an open set, there are numbers a and b with a < b where x is in (a,b) and (a,b) is a subset of U. From this we have (a,b) is a subset of Z. The case when x is in V is handled similarly. Thus Z is an open set. EOP.

1. Are the statements in propositions 1 and 2 conditional or absolute? If conditional, what are the hypotheses and conclusions? If absolute, what is the main quantifier?

2. List the variables used in these propositions. Indicate what these variables represent.

3. The proofs of these propositions are direct. For each proof : 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 (goal) is altered.

4. Did the proofs explicitly or implicitly leave some steps for the reader to complete? If so, state what steps the reader is expected to complete. [Optional: complete these steps.]

5. 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.]

6. Overall, do you think these proofs were effective? Discuss briefly the basis for you conclusion.