My Math Forum  

Go Back   My Math Forum > College Math Forum > Applied Math

Applied Math Applied Math Forum


Thanks Tree2Thanks
  • 1 Post By jenniferruurs
  • 1 Post By jenniferruurs
Reply
 
LinkBack Thread Tools Display Modes
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.
jenniferruurs is offline  
 
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 .
idontknow is offline  
March 15th, 2019, 02:30 PM   #3
Newbie
 
Joined: Dec 2018
From: Amsterdam

Posts: 28
Thanks: 2

Quote:
Originally Posted by idontknow View Post
Simply there is another symbol of relation R which is an opposite of the given relation .
What do you mean with this?
Thanks from idontknow
jenniferruurs is offline  
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 .
idontknow is offline  
March 15th, 2019, 11:49 PM   #5
Newbie
 
Joined: Dec 2018
From: Amsterdam

Posts: 28
Thanks: 2

Quote:
Originally Posted by idontknow View Post
$\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))))?
jenniferruurs is offline  
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 .
idontknow is offline  
March 16th, 2019, 08:23 AM   #7
Newbie
 
Joined: Dec 2018
From: Amsterdam

Posts: 28
Thanks: 2

Quote:
Originally Posted by idontknow View Post
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?
Thanks from idontknow
jenniferruurs is offline  
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.
idontknow is offline  
Reply

  My Math Forum > College Math Forum > Applied Math

Tags
form, formulas, opposite, predicate, rewrite



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
opposite concepts shaharhada Calculus 2 December 16th, 2016 07:29 AM
Rewrite an expression in a specific form apartin Calculus 0 March 10th, 2016 01:17 PM
Opposite numbers deepblue Elementary Math 5 June 9th, 2015 06:44 AM
How to rewrite mathkid Calculus 2 September 16th, 2012 04:11 PM
What is the opposite of unitless? Scrimski Algebra 5 October 10th, 2008 04:57 AM





Copyright © 2019 My Math Forum. All rights reserved.