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

[sktsasus]

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]



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[tkhunny]

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

Let's see your first attempts.

Oh, and probably this one: p implies q is equivalent to (~p) OR q.


No one mentioned all this in class?