Math 240 Proof Evaluation #6 Spring,1999
M. Flashman Due: 32499
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, A_{a
}is a set .
Then we define the intersection of the family A_{a}
for a in
I by
Ç
A_{a
}= {x : x is a member of A_{a
}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 A_{a}
is an open set of real numbers for every a
in I, then Ç
A_{a}
is an open set.
Proof: [This
proof is erroneous.]
Suppose x is a member
of Ç
A_{a}
. Then for every b
in I, x is a member of A_{b}.
Since A_{b
}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_{b}
for every b
in I, and hence (a,b) is a subset of Ç
A_{a
}_{. }Therefore_{ }Ç
A_{a
}is an open set. EOP.

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?

Are the proofs of these propositions
direct or indirect?

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

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.

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.

The proof of Proposition 2 has
an error in it. Describe any errors you find in this "proof" of proposition
2.

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.]

Generalize Proposition 1 and
give a proof for your generalization.

Overall, do you think the proof
of proposition 1 was effective? Discuss briefly the basis for you conclusion.