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 March 15th, 2019, 07:37 AM #1 Newbie   Joined: Dec 2018 From: Amsterdam Posts: 28 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: 645 Thanks: 92 Simply there is another symbol of relation R which is an opposite of the given relation . March 15th, 2019, 02:30 PM   #3
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 Originally Posted by idontknow Simply there is another symbol of relation R which is an opposite of the given relation .
What do you mean with this? March 15th, 2019, 02:54 PM #4 Senior Member   Joined: Dec 2015 From: somewhere Posts: 645 Thanks: 92 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
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 Originally Posted by idontknow $\displaystyle \neg A =\forall x \not\exists y (xRy)\;$ , 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))))? March 16th, 2019, 08:10 AM #6 Senior Member   Joined: Dec 2015 From: somewhere Posts: 645 Thanks: 92 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
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 Originally Posted by 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 .
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? March 16th, 2019, 08:42 AM #8 Senior Member   Joined: Dec 2015 From: somewhere Posts: 645 Thanks: 92 Yes this way gives the full or true opposite , removing the implication. Tags form, formulas, opposite, predicate, rewrite Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post shaharhada Calculus 2 December 16th, 2016 07:29 AM apartin Calculus 0 March 10th, 2016 01:17 PM deepblue Elementary Math 5 June 9th, 2015 06:44 AM mathkid Calculus 2 September 16th, 2012 04:11 PM Scrimski Algebra 5 October 10th, 2008 04:57 AM

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