0

Ok, I have the given relation: If F(x) is not true then no case satisfies G(x) and H(y,x). ((∀x ¬F(x)) ⇒¬(∀y G(y) ˄ H(y,x)))

Now, Can I possibly convert this into: (∀y G(y) ˄ H(y,x))) ⇒ ((∀x F(x)) ????

If not, the left hand side essentially has to imply: If F(x) is not true.... Mentions nothing about the For All or Existential Quantifiers. Can I take the negation outside of the Quantifier i.e. put it as (¬(∀x F(x)), because this makes the job much easier???

Volker Stolz
  • 7,274
  • 1
  • 32
  • 50
gran_profaci
  • 8,087
  • 15
  • 66
  • 99

1 Answers1

2

I'm not sure this is the right place but, no you can't. Moving the negation out would change the quantifier. Also, the initial formula may not be what you want: the last x is a free variable.

NotAUser
  • 1,436
  • 8
  • 12
  • I see, what if this was the statement now?(∀x ¬F(x)) ⇒¬(∃ y G(y)) ˄ H(y,x))? Sorry, I made an error before... – gran_profaci Sep 26 '12 at 19:48
  • All you need here is two basic equivalences. Apologies for the notation: (1) ∀x ¬F = ¬ ∃x F (2) F ⇒ G = ¬ F v G. You can easily find out the answer by yourself :) – NotAUser Sep 27 '12 at 15:13