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March 15th, 2019, 07:37 AM  #1 
Newbie Joined: Dec 2018 From: Amsterdam Posts: 26 Thanks: 2  How to rewrite these formulas in the opposite form?
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 skipjack; March 15th, 2019 at 01:30 PM. 
March 15th, 2019, 08:22 AM  #2 
Senior Member Joined: Dec 2015 From: somewhere Posts: 511 Thanks: 79 
Simply there is another symbol of relation R which is an opposite of the given relation .

March 15th, 2019, 02:30 PM  #3 
Newbie Joined: Dec 2018 From: Amsterdam Posts: 26 Thanks: 2  
March 15th, 2019, 02:54 PM  #4 
Senior Member Joined: Dec 2015 From: somewhere Posts: 511 Thanks: 79 
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 . 
March 15th, 2019, 11:49 PM  #5  
Newbie Joined: Dec 2018 From: Amsterdam Posts: 26 Thanks: 2  Quote:
∀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))))?  
March 16th, 2019, 08:10 AM  #6 
Senior Member Joined: Dec 2015 From: somewhere Posts: 511 Thanks: 79 
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 . 
March 16th, 2019, 08:23 AM  #7  
Newbie Joined: Dec 2018 From: Amsterdam Posts: 26 Thanks: 2  Quote:
∃w1w2w3(w1Rw2 ∧ w1Rw3 ∧ (not ∃w(not exist w)(w2Rw ∧ w3Rw)))) This is the correct formula do you agree?  
March 16th, 2019, 08:42 AM  #8 
Senior Member Joined: Dec 2015 From: somewhere Posts: 511 Thanks: 79 
Yes this way gives the full or true opposite , removing the implication.


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