Delete.
LEQV = Logically equivalent
I need help changing
∀xA(x) ^ ∀xB(x) ^ ∀xC(x) → ∀xD(x)
into PNF (such that the only connectives in the quantifier free portion have to be →)
∀xA(x) ^ ∀xB(x) ^ ∀xC(x) → ∀xD(x)
LEQV
￢ ( ∀xA(x) ^ ∀xB(x) ^ ∀xC(x) ) v ∀xD(x) [→ Law]
LEQV
( ∃x￢A(x) v ∃x￢B(x) v ∃x￢C(x) ) v ∀xD(x) [Duality of quantifiers law]
LEQV
∃x1∃x2∃x3∀x ( ( ￢A(x1) v ￢B(x2) v ￢C(x3) ) v D(x) ) [pulling quantifiers out]
LEQV
∃x1∃x2∃x3∀x ( ￢ ( A(x1) ^ B(x2) ^ C(x3) ) v D(x) ) [pulling￢ out]
LEQV
∃x1∃x2∃x3∀x ( ( A(x1) ^ B(x2) ^ C(x3) ) → D(x) ) [reverse → Law ]
I got here but I need the only connectives in the quantifier free portion to be →, right now the last step includes ^ symbols...
Thanks
Hmm is this right?
∀xA(x) ^ ∀xB(x) ^ ∀xC(x) → ∀xD(x)
LEQV
￢ ( ∀xA(x) ^ ∀xB(x) ^ ∀xC(x) ) v ∀xD(x) [→ Law]
LEQV
∃x￢A(x) v ∃x￢B(x) v ∃x￢C(x) v ∀xD(x) [Duality of quantifiers law]
LEQV
∃x1∃x2∃x3∀x ( ￢A(x1) v ￢B(x2) v ￢C(x3) v D(x) ) [pulling quantifiers out]
LEQV
∃x1∃x2∃x3∀x ( A(x1) → (￢B(x2) v ￢C(x3) v D(x) ) ) [reverse → Law ]
LEQV
∃x1∃x2∃x3∀x ( A(x1) →( B(x2) →( ￢C(x3) v D(x) )) ) [reverse → Law ]
LEQV
∃x1∃x2∃x3∀x ( A(x1) →( (B(x2) → (C(x3) → D(x) )) ) [reverse → Law ]
Unless I'm missing something (and I'll use 'V' and 'E' as the quantifiers), we have:
(VyAy & VzBz & VwCw) -> VxDx
equivalent to
VxEyzw(Ay -> (Bz -> (Cw -> Dx)))
But I don't see how you would get
(VyAy & VzBz & VwCw) -> VxDx
equivalent to
EyzwVx(Ay -> (Bz -> (Cw -> Dx)))
I think I was correct when I said (before edits) that you reversed the correct order of quantifiers when you "pulled out".
I don't see how this is wrong
∃x1∃x2∃x3∀x ( ￢A(x1) v ￢B(x2) v ￢C(x3) v D(x) ) [pulling quantifiers out]
because when I pull out the quantifiers, I renamed the variables using this law
Discrete Structures, Logic, and ... - Google Books
so each variable in the quantifiers are different, so the order shouldn't matter right?
Suppose and are quantifiers (∀ or ∃) and is ∧ or ∨. Then
is equivalent to both
and
(assuming x does not occur freely in B and y does not occur freely in A). Therefore, in this problem the order of quantifiers in the prenex form does not matter.