Question

How to rewrite ~∀x((∀y∀z P(x,y,z))-->(∃z∃y R(x,y,z))) so that the negations appear only within prediates?

Answer 1

That is no negation is outside a quantifier or an expression involving logical connectives.

I don't see how you can do that? Is it asking how to get the NOT in front of the P? I tried using distributive law but now I'm just confusing myself.. Any help would be great. If you could list the laws you use that would be great as well I actually want to understand this

Answer 2

I posted this earlier but didn't get a response any ideas would be very much appreciated

Answer 3

I have noticed many times that when you have to puch a negative thru, it tends to flip the operator

Answer 4

-Ax P(x) = Ex -P(x)

Answer 5

\[-\int_{a}^{b}f(x)dx=\int_{b}^{a}f(x)dx\]

Answer 6

\[-(P\ \cap\ Q )=(-P\ \cup\ -Q )\]

Answer 7

I tried that before but I only got this far ∃x ~((∀y∀z P(x,y,z))

Answer 8

then didn't know how to proceed from there I think the question wants me to get the ~ infront of the P

Answer 9

change the innards from p->q to its equivalent -p v q

Answer 10

then a negation across it all will look more doable

Answer 11

Okay I got ∃x~(~(∀y∀z P(x,y,z)) v (∃z∃y R(x,y,z)))

Answer 12

can I multiply in the ~ using distributive law?

Answer 13

∃x((∀y∀z P(x,y,z)) ^ ~(∃z∃y R(x,y,z)))

Answer 14

\[-∀x((∀y∀z P(x,y,z))\rightarrow(∃z∃y R(x,y,z)))\]
\[-∀x(-(∀y∀z P(x,y,z))\cup\ (∃z∃y R(x,y,z)))\]
\[Ex\ -(-(∀y∀z P(x,y,z))\cup\ (∃z∃y R(x,y,z)))\]
\[Ex\ (--(∀y∀z P(x,y,z))-\cup\ -(∃z∃y R(x,y,z)))\]
\[Ex\ ((∀y∀z P(x,y,z))\cap\ (-∃z∃y R(x,y,z)))\]
\[Ex\ ((∀y∀z P(x,y,z))\cap\ (Az-∃y R(x,y,z)))\]
\[Ex\ ((∀y∀z P(x,y,z))\cap\ (AzAy -R(x,y,z)))\]
perhaps

Answer 15

Oh snap is that a union sign?

Answer 16

and and ors :)

Answer 17

? the U and then the upside down U

Answer 18

yeah, they come out better in the latex code when trying to do ^ and v

Answer 19

same directions, same meaning, just rounded ends

Answer 20

Ohhh I see so the U is or and the upsidedown U is and?

Answer 21

yep

Answer 22

sorry I started learning about U signs to in class about sets or something so I got confused thanks again!

Answer 23

your welcome