Math 240                 Proof Evaluation #5
More on Open Sets of Real Numbers
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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 F is a family of sets. We define the intersection of the family of sets F by

Ç F = {x : x is a member of A for every A in the family F} .

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 F is a  family and any member of F is open set of real numbers, then Ç F is an open set.

Proof: [This proof is erroneous.]
Suppose x is a member of Ç F . Then  x is a member of A for every A in F. Since A 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 A for every A in F, and hence (a,b) is a subset of Ç F . Therefore Ç F is an open set. EOP.

Proposition 3:
For n a positive integer, let  An = (5-1/n, 5 + 1/n) and let F = {An : n a positive integer}.  Ç F= {5}.

Proof:
If x is not 5, Then  | x - 5 |  > 0, and there is an natural number  n where  | x - 5 |  > 1/n >0 so that x is not an element of  An. Thus by the definition  of  Ç F, x not a member of Ç F. Thus by the contrapositive, if x is a member of  Ç F , then = 5. So Ç F  is a subset of {5}.  Clearly, 5 is an element of An = (5-1/n, 5 + 1/n)  for all n , so {5} is a subset of  Ç F,  and thus  Ç F = {5}.
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. Proposition 2 is false. Why does Proposition 3 show that Proposition 2 is false.
3. The proof of Proposition 2 has an error in it. Describe any errors you find in this "proof" of proposition 2.
4. 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.]
5. For n a positive integer, let  Bn = (4-1/n, 6 + 1/n) and let F = {Bn : n a positive integer}.
Prove:
Ç F is the closed interval of real numbers [4,6].
6. True or false: The intersection of  a family of open sets of real numbers is open. Discuss your response.