# How to rewrite these formulas in the opposite form?

#### jenniferruurs

I want to know how I can write from some predicate formula its negative form.

For example
âˆ€xâˆƒy(x R y)
then not(âˆ€xâˆƒy(x R y)) is equal to âˆƒx,not âˆƒy(x R y)))(not exist y)

How do these formula work with conjunction implication and disjunction?

Bellow are examples of some formulas but I also want to know how to rewrite the formulas if there are conjunctions and disjunctions involved.

$\forall a,b,c \in X: (aRb \wedge bRc) \Rightarrow aRc$
what do we have if we have the negation of this formula?

also the opposite of:
$\forall a, b \in X(a R b \Rightarrow \lnot(b R a))$

also for the opposite of:
$\forall a, b, c: a R b \land b R c \Rightarrow \lnot (a R c)$

also for the opposite of
$\forall a, b, c\in X\,(a\,R\, b \land a \,R\, c \to b \,R\, c)$

also for the opposite of:
$\forall a, b \in X(a R b \Leftrightarrow b R a)$

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#### idontknow

Simply there is another symbol of relation R which is an opposite of the given relation .

#### jenniferruurs

Simply there is another symbol of relation R which is an opposite of the given relation .
What do you mean with this?

• 1 person

#### idontknow

About the example $$\displaystyle A : \forall x \exists y (x R y)$$ we can have two opposites of A .
$$\displaystyle \neg A =\forall x \not\exists y (xRy)\;$$ , but instead of $$\displaystyle \forall x$$ the $$\displaystyle \exists ! x$$ ( exists only one x) also is a type of opposite of A .
So the best way is to just simply write the opposite of relation R with a symbol .

#### jenniferruurs

$$\displaystyle \neg A =\forall x \not\exists y (xRy)\;$$ , but instead of $$\displaystyle \forall x$$ the $$\displaystyle \exists ! x$$ ( exists only one x) also is a type of opposite of A .
So the best way is to just simply write the opposite of relation R with a symbol .
How would you rewrite this
âˆ€w1w2w3(w1Rw2 âˆ§ w1Rw3 â†’ âˆƒw(w2Rw âˆ§ w3Rw))

If you want to rewrite it like:
not(âˆ€w1w2w3(w1Rw2 âˆ§ w1Rw3 â†’ âˆƒw(w2Rw âˆ§ w3Rw))))

Can this be rewritten as:
âˆƒw1w2w3(w1Rw2 âˆ§ w1Rw3 â†’ (not âˆƒw(not exist w)(w2Rw âˆ§ w3Rw))))?

#### idontknow

Yes but still $$\displaystyle \rightarrow \exists$$ can have the opposite like :
$$\displaystyle \rightarrow \not\exists$$ and $$\displaystyle \not\rightarrow \exists$$ . So you can write the opposite but depending on what relation is .

#### jenniferruurs

Yes but still $$\displaystyle \rightarrow \exists$$ can have the opposite like :
$$\displaystyle \rightarrow \not\exists$$ and $$\displaystyle \not\rightarrow \exists$$ . So you can write the opposite but depending on what relation is .
I think that you and I had it both wrong and that the implication had to be removed.

âˆƒw1w2w3(w1Rw2 âˆ§ w1Rw3 âˆ§ (not âˆƒw(not exist w)(w2Rw âˆ§ w3Rw))))

This is the correct formula do you agree?

• 1 person

#### idontknow

Yes this way gives the full or true opposite , removing the implication.