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:

$\forall a, b \in X(a R b \Leftrightarrow b R a)$

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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