M. Flashman **Due: 2-24-99**

More on Open Sets of Real Numbers

Reminder of the Definitions:

(1) For a and 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 some numbers a and b so
that x is a member of (a,b) and (a,b) is a subset of O.

**Proposition:** The empty set is an open set.

**Proof 1**: Suppose N is not an open set. Then there is some number
x that is a member of N and for any numbers a and b with x a member of
(a,b), the set (a,b) is not a subset of N. Thus if N is not an open set,
N is not the empty set. Therefore ( by the contrapositive) the empty set
is an open set. EOP.

**Proof 2**: Consider the open sets (0,1) and (3,4). We see that
(0,1) (3,4) is the empty set. But the intersection of two open sets is
an open set. Therefore the empty set is an open set. EOP.

1. Is the statement in the proposition conditional or absolute? If conditional, what are the hypotheses and conclusions? If absolute, can you rephrase the statement as a conditional statement?

2. Are the proofs of this proposition direct or indirect?

a. If the proof is indirect, state the way in which the argument proceeds.(What is assumed? What is actually demonstrated?)

b. 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.

3. 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.]

4. 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.]

5. Indicate any parts of the argument that you felt needed greater detail or better connection to the proofs. [Optional: Supply these details or suggest a better connection.]

6. Consider {x: 0<x <= 1}= (0,1]. Prove: (0,1] is not an open set. [Optional: Generalize this result. Prove your generalization.]

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