1. ## negating the expression

hey guys I have this question: Negatate.

$\exists x \forall y (( y=x-7) \vee (x < 2y))$

$
\Rightarrow \forall x \exists y \sim(( y=x-7) \vee (x < 2y))$

$\Rightarrow \forall x \exists y ( \sim ( y=x-7) \wedge \sim(x < 2y))$

$\Rightarrow \forall x \exists y ( ( y \neq x-7) \wedge (x \geq 2y))$

This this OK? thanks...

2. It seems well done to me.
That is the way I would do it.

3. What's the definition of "negate F" where 'F' is a formula?

If the definition of "negate F" is "show the negation of F", then the answer is:

~F

A formula F has exactly one formula that is the negation of F, and that formula is ~F.

Other formulas may be equivalents of the negation of F, but only ~F is the negation of F.

Your instructor or text needs to provide a precise definition of "negate F", if what is intended is something other than "show the negation of F".

/

Also, in your answer you assumed that "not less than" holds if and only if "greater than or equal" holds. But that we can derive "(not less than) iff and only if (greater than or equal)" is specific only to certain theories (it is not a theorem of logic alone); and "(not less than) if and only if (greater than or equal)" is true only in certain interpretations of the language (it is not a logical truth).

Otherwise, the problem should have included something such as, "Where the formulas are interpreted for real numbers" or something like that.

4. Originally Posted by Plato
It seems well done to me.
That is the way I would do it.
sorry is the negation of less than.... greater than? or greater than and equal too?

5. In the universe of the real numbers $\neg \left( {x < y} \right) \equiv x \geqslant y$.