Math negation equation

  • Thread starter gnome
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  • #1
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I want to negate this: [itex]( \exists x ) ( \forall y ) \Phi (x,y ) [/itex]

Is this correct?

[tex]\neg ( ( \exists x ) ( \forall y ) \Phi (x,y ) ) \equiv ( \forall x ) ( \exists y ) \neg \Phi (x,y ) [/tex]
 

Answers and Replies

  • #2
AKG
Science Advisor
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looks right
 
  • #3
gnome said:
I want to negate this: [itex]( \exists x ) ( \forall y ) \Phi (x,y ) [/itex]

Is this correct?

[tex]\neg ( ( \exists x ) ( \forall y ) \Phi (x,y ) ) \equiv ( \forall x ) ( \exists y ) \neg \Phi (x,y ) [/tex]
Yes.

~[(Ex)(Ay)F(x,y)] <-> ~(Ex)(Ay)F(x.y)
~(Ex)(Ay)F(x,y) <-> (Ax)~(Ay)F(x,y)
(Ax)~(Ay)F(x,y) <-> (Ax)(Ey)~F(x,y)
therefore,
~[(Ex)(Ay)F(x,y)] <-> (Ax)(Ey)~F(x,y).
 
  • #4
1,036
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Thanks guys.
 

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