# Math Help - Negation help (2)

1. ## Negation help (2)

Question, Negate:

$\exists x \big{(}\big{(}(xPx) \wedge \sim (xQx)\big{)} \wedge \forall y \big{(}(xQy) \wedge \big{(} \sim (yPy) \wedge \sim (yQx)\big{)}\big{)}\big{)}$

$\Rightarrow$ $\sim (\exists x \big{(}\big{(}(xPx) \wedge \sim (xQx)\big{)} \wedge \forall y \big{(}(xQy) \wedge \big{(} \sim (yPy) \wedge \sim (yQx)\big{)}\big{)}\big{)}\big{)}$

$\Rightarrow$ $\forall x \sim \big{(}\big{(}(xPx) \wedge \sim (xQx)\big{)} \wedge \forall y \big{(}(xQy) \wedge \big{(} \sim (yPy) \wedge \sim (yQx)\big{)}\big{)}\big{)}$

$\Rightarrow$ $\forall x \big{(} \sim \big{(}(xPx) \wedge \sim (xQx)\big{)} \vee \sim \forall y \big{(}(xQy) \wedge \big{(} \sim (yPy) \wedge \sim (yQx)\big{)}\big{)}\big{)}$

$\Rightarrow$ $\forall x \big{(} \big{(} \sim (xPx) \vee (xQx)\big{)} \vee \exists y \big{(} \sim (xQy) \vee \sim \big{(} \sim (yPy) \wedge \sim (yQx)\big{)}\big{)}\big{)}$

$\Rightarrow$ $\forall x \big{(} \big{(} \sim (xPx) \vee (xQx)\big{)} \vee \exists y \big{(} \sim (xQy) \vee \big{(} (yPy) \vee (yQx)\big{)}\big{)}\big{)}$

Is this correct? thank you for your help.

2. Can anyone confirm? thank you

3. I don't see that you negated the original formula. What I see is that you listed equivalents of the original formula.

The negation of a formula F is the formula ~F.

So the negation of

Ex(xPx & ~xQx & Ay(xQy & ~yPy & ~yQx))

is

~Ex(xPx & ~xQx & Ay(xQy & ~yPy & ~yQx))

4. Originally Posted by MoeBlee
I don't see that you negated the original formula. What I see is that you listed equivalents of the original formula.

The negation of a formula F is the formula ~F.

So the negation of

Ex(xPx & ~xQx & Ay(xQy & ~yPy & ~yQx))

is

~Ex(xPx & ~xQx & Ay(xQy & ~yPy & ~yQx))
what you wrote was my first line after that i kept nagating everything inside the brackets with the use of negation laws.

Your also missing many other brackets.

5. Originally Posted by jvignacio
what you wrote was my first line :) after that i kept nagating everything inside the brackets with the use of negation laws.

Your also missing many other brackets.
Oops, sorry I missed that you did negate at your first line.

As to parentheses, I just took the informality of eliminating a lot unnecessary parentheses.

But there's a technical point (pedantic but nevertheless it is best to be precise) (and it might be a problem with the wording of the problem itself): The negation of a formula F is ~F. So when asked to negate a formula, we don't have to go on also to mention other equivalents derived by such things as DeMorgan's laws and quantifier exchange. In fact, those extra equivalents are NOT the negation of your original formula. They are EQUIVALENTS of the negation but are not themselves the negation.

That is to say, if I answered that question on a test, I would just put the negation sign on the outside of the formula and move on to the next question. If the professor said that I'm wrong because I didn't transform further, then I'd tell him that he's wrong. The negation of a formula F is ~F period, and any other transformation gives an EQUIVALENT of the negation but not the negation ITSELF. If asked to negate F, the correct answer is to write ~F.

6. Originally Posted by MoeBlee
Oops, sorry I missed that you did negate at your first line.

As to parentheses, I just took the informality of eliminating a lot unnecessary parentheses.

But there's a technical point (pedantic but nevertheless it is best to be precise) (and it might be a problem with the wording of the problem itself): The negation of a formula F is ~F. So when asked to negate a formula, we don't have to go on also to mention other equivalents derived by such things as DeMorgan's laws and quantifier exchange. In fact, those extra equivalents are NOT the negation of your original formula. They are EQUIVALENTS of the negation but are not themselves the negation.

That is to say, if I answered that question on a test, I would just put the negation sign on the outside of the formula and move on to the next question. If the professor said that I'm wrong because I didn't transform further, then I'd tell him that he's wrong. The negation of a formula F is ~F period, and any other transformation gives an EQUIVALENT of the negation but not the negation ITSELF. If asked to negate F, the correct answer is to write ~F.
wow! in class we were taught to negate further until we cant negate anymore. I don't think if I have the guts to just add a negation sign at the front of the question and move on. I know according to you its OK too but since we were taught the lecturers way, I guess I should answer the lecturers way. Thanks you anyway for helping out and explaining

7. in class we were taught to negate further until we cant negate anymore.
That is misstated. If that is what your professor said, then he is using sloppy terminology. First, you negated, then you didn't continue to negate, rather you showed (through various laws such as DeMorgan's and quantifier exchange) successive formulas equivalent to the negated formula.

If the problem is to negate the formula and then show an equivalent formula that has the negation sign only in front of atomic formulas, then the last line in your list is the answer. But the problem was just to negate the formula, so the first formula in your list is the answer.

I guess I should answer the lecturers way. Thanks you anyway for helping out and explaining
As far as intellectual integrity is concerned, you should answer the question correctly. If the lecturer has misstated the problem, then you should tell him that. As for getting a good grade in the class, though, probably you should answer as your lecturer would want you to.

So, if this is just another class to get a grade in, then it doesn't much matter that the lecturer has presented the problem incorrectly. But if you are seriously interested in the subject of logic, then you should be clear that the negation of a formula F is simply ~F, and that further equivalents of ~F are not negations of ~F nor are they negations of F. Rather they are EQUIVALENTS of the negation of F. There is only ONE formula that is the negation of F and that is ~F, period.

I know according to you its OK
It's not a matter of what according to me is okay. It's a matter of what is correct. It simply is incorrect to say that anything other than ~F is the negation of F (where '~' is whatever negation sign is in the language; or it could be ~(F) if the formation rules of the system dictate parentheses in that manner).