Let be an open sentence over a domain .

is a false statement and that the set of counterexamples is a proper subset of .

If I let be true and be false, is

a correct expression?

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- Mar 4th 2010, 03:02 PMnoviceLogical expressions
Let be an open sentence over a domain .

is a false statement and that the set of counterexamples is a proper subset of .

If I let be true and be false, is

a correct expression? - Mar 4th 2010, 03:22 PMPlato
- Mar 4th 2010, 05:37 PMnovice
Exercise 5.34, from Mathematical Proofs by Chartrand, page 125.

Let R(x) be an open sentence over a domain . Suppose that is a false statement and that the set T of counterexamples is a proper subset. Show that there exists a nonempty subset W of S such that is true.

My attempt:

Let be true and be false. Then

Since , it follows that .

The negation of the statment is

or equivalently

, where

I don't know how to get is true. - Mar 5th 2010, 08:16 AMPlato
I think that is self-contradictory problem.

How can an open sentence be false ?

And then true on some nonempty subset of .

I don’t think I have ever seen that textbook. There may be some way it is using the expression “the set T of counterexamples is a proper subset ” .

I find that idea a bit odd. - Mar 5th 2010, 08:56 AMnovice
- Mar 5th 2010, 10:01 AMemakarovQuote:

Usually, is not considered a well-formed formula, though it makes sense to view it as an abbreviation for . In fact, I don't think " " part is needed in the formula at all.

Concerning semantics, this formula is true according to the description of , and .

Quote:

The negation of the statment is

The formula does not follow from , or its negation, alone. The latter formula says that elements of make false, but it does not say anything about elements outside .

On the other hand, does imply (in fact, the implication to the left is enough). - Mar 5th 2010, 06:37 PMnovice

Thank you for pointing at the well formed formula. I noticed that my math book has violated wff left and right. I read a book on logic over a Christmas holiday out of curiosity. I finished reading the entire book in two weeks. Ha ha, now you know what quality education I obtained in logic.