Math 240 Proof Evaluation #6 Spring,1999
M. Flashman Due: 3-24-99
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.
(3) Suppose I is a set and for each a in I, Aa is a set . 
Then we define the intersection of the family Aa for a in I by

Ç Aa = {x : x is a member of Aa for every a in the set I} .

Proposition 1 : {5} is not an open set.
Proof: Suppose {5} is an open set.
Consider the number 5, which is an element (in fact the only element) of {5}. Suppose a and b are any real numbers, where a < 5 < b. Then a < (5+a)/2 < 5 and therefore (a,b) is not a subset of {5}. Thus {5} is not an open set. EOP.

Proposition 2: [This proposition is FALSE.]
If Aa is an open set of real numbers for every a in I, then Ç Aa is an open set.

Proof: [This proof is erroneous.]
Suppose x is a member of Ç Aa . Then for every b in I, x is a member of Ab. Since Ab is an open set, there are real numbers a and b where x is a member of (a,b) and (a,b) is a subset of Ab for every b in I, and hence (a,b) is a subset of Ç Aa . Therefore Ç Aa is an open set. EOP.

  1. Are the statements in the propositions 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 these propositions direct or indirect?
    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.
  3. Proposition 2 is false. Construct an example of a family of open sets so that the interesection of the family is {5}. Why does your example show that proposition 2 is false.
  4. The proof of Proposition 2 has an error in it. Describe any errors you find in this "proof" of proposition 2.
  5. Indicate any parts of the argument in proposition 1 that you felt needed greater detail or better connection. [Optional: Supply these detail or suggest a better connection.]
  6. Generalize Proposition 1 and give a proof for your generalization.
  7. Overall, do you think the proof of proposition 1 was effective? Discuss briefly the basis for you conclusion.