How to rewrite these formulas in the opposite form?

Dec 2018
42
2
Amsterdam
I want to know how I can write from some predicate formula its negative form.

For example
∀x∃y(x R y)
then not(∀x∃y(x R y)) is equal to ∃x,not ∃y(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:
$\forall a, b \in X(a R b \Leftrightarrow b R a)$
 
Last edited by a moderator:
Dec 2015
1,082
169
Earth
Simply there is another symbol of relation R which is an opposite of the given relation .
 
Dec 2015
1,082
169
Earth
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 .
 
Dec 2018
42
2
Amsterdam
\(\displaystyle \neg A =\forall x \not\exists y (xRy)\; \) , 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
∀w1w2w3(w1Rw2 ∧ w1Rw3 → ∃w(w2Rw ∧ w3Rw))

If you want to rewrite it like:
not(∀w1w2w3(w1Rw2 ∧ w1Rw3 → ∃w(w2Rw ∧ w3Rw))))

Can this be rewritten as:
∃w1w2w3(w1Rw2 ∧ w1Rw3 → (not ∃w(not exist w)(w2Rw ∧ w3Rw))))?
 
Dec 2015
1,082
169
Earth
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 .
 
Dec 2018
42
2
Amsterdam
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.

∃w1w2w3(w1Rw2 ∧ w1Rw3 ∧ (not ∃w(not exist w)(w2Rw ∧ w3Rw))))

This is the correct formula do you agree?
 
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Dec 2015
1,082
169
Earth
Yes this way gives the full or true opposite , removing the implication.