# How would I match up the following sentences with their respective negations?

#### sktsasus

##### New member
A) ∀x(P(x) ∧ [(∃yP(y)) ∧ (∃yR(x, y)) ⇒ (∃yP(y))])
B) ∀x(P(x) ⇔ ∃yR(x, y))

1) ∃x([P(x) ∨ ∃yR(x, y)] ∧ [¬P(x) ∨ ∀y¬R(x, y)])
2) ∃x¬P(x)

A and B are the sentences and 1 and 2 are the negations. I first tried to get the negation of B and this is how far I got:
[FONT=MathJax_Main]
I did the quantifier negation first and then the next steps:

¬
[/FONT][FONT=MathJax_Main]∀[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]P[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main][/FONT][FONT=MathJax_Main]∃[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Math-italic]R[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main])
[/FONT][FONT=MathJax_Main]= ∃[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]¬[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]P[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main][/FONT][FONT=MathJax_Main]∃[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Math-italic]R[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main])
[/FONT][FONT=MathJax_Main]= ∃[FONT=MathJax_Math-italic]x[FONT=MathJax_Main]¬[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]P[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]∧[/FONT][FONT=MathJax_Main]∃[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Math-italic]R[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]∨[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]¬[/FONT][FONT=MathJax_Math-italic]P[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]∧[/FONT][FONT=MathJax_Main]¬[/FONT][FONT=MathJax_Main]∃[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Math-italic]R[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main])

Any help on how I could proceed? [/FONT][/FONT][/FONT]

Last edited:

#### tkhunny

##### Moderator
Staff member
Well, there will be a lot of DeMorgan in your future. That can be a good place to start.

Also these:

Double Negative Law
Commutative Law for conjunction.
Commutative Law for disjunction.
Associative Law for conjunction.
Associative Law for disjunction.
Distributive Laws
Absorption Laws