Question

Provide a proof using standard predicate logic (without using the deduction method, proof by contradiction, or proof by contraposition) for the validity of the following argument:

(∀x)A(x) ∧ (∃x)B(x) → (∃x)[A(x) ∧ B(x)]

Answer 1

I'm trying to solve this now. If anyone wants to work with me then that would be great; otherwise, I have no way to check my proof.

Answer 2

If you care to show your solution I am happy to read it for you. What I remember from my course (which was very long ago) was to in order to show such statements we had to use axioms, in fact we had 13. It is indeed hard to judge if you're working with the same axiom system as I did.

If you work with an axiom system you need to share your axioms in order to get help. Aside from that, I can only tell that the implication is indeed true. Because if a Formula A(x) is true for all its arguments and there exists an x such that B(x) is true then of course there exists an x such that A(x) and B(x) are true.

This is however far from a formal proof. If you need a formal one, we need your axioms.

Answer 3

@Spacelimbus

Answer 4

@Spacelimbus Thank you for replying. I like your informal proof :) I am still working on trying to figure this out.

Answer 5

@zepdrix how are you with predicate logic?

Answer 6

Ok sorry sorry was finishing up a game :d
What's going on now?

Answer 7

If `For all x, A(x)` `and` `there exists an x such that B(x)`,
then `there exists an x such that` `A(x) and B(x)`.

This is really weird stuff 0_o
I don't remember ever doing this nonsense with these quantifiers.. hmm

Answer 8

The good news is-- I'm fine with induction and propositional logic now... this is the last piece of the puzzle.

Answer 9

what game were you playing?

Answer 10

Rocket League XD lol

Answer 11

Had to give up the addicting games so I could really really focus on schooling this semester :P

Rocket League is mindless fun though, nothing that I can't let go of.

Answer 12

that's cray. i've never seen a vehicle based soccer game before, lol

Answer 13

i only play chess and starcraft :X

Answer 14

Hmm I gotta think about this problem a bit :O

Answer 15

Ummm ok let's see if this works or not...

Answer 16

The first piece: \(\large\rm \forall x, A(x)\)
For all x, A(x).

So any x's that we have, all of them, satify this statement A.
So all together we have the union of a bunch of statements A involving different x's.
\[\large\rm \forall x, A(x)\quad\equiv\quad A(x_1)\cup A(x_2)\cup...\cup A(x_n)\]

Answer 17

This next piece: \(\large\rm \exists x: B(x)\)
Which translates to `There exists an x such that B(x)`,
tells us that there is a single x in existence that satisfies the statement B.
Let's call it x_i, just one of the many x's we could list.
\[\large\rm \exists x: B(x)\quad\equiv\quad B(x_i)\]

Answer 18

And of course, the caret shape is to denote "and" or "intersection.

Answer 19

So on the left side of our thing we have,\[\large\rm \left[A(x_1)\cup A(x_2)\cup...\cup A(x_n)\right]\cap B(x_i)\]

Answer 20

They only intersect at the same x,\[\large\rm A(x_i)\cap B(x_i)\]And we need to see if that `implies` the right part.

I don't think I'm doing this right -_- it's been too long...

Answer 21

On the right side, \(\large\rm \exists x:\left[A(x)\wedge B(x)\right]\)
Translates to:
There exists an x such that, both A(x) and B(x).
Again, let's call this x, x_i.
So the right side says that some x exists, so that the intersection of A(x) and B(x) is satisfied.\[\large\rm \exists x:\left[A(x)\wedge B(x)\right]\quad\equiv\quad A(x_i)\cap B(x_i)\]

Answer 22

But again, I don't think I'm doing this correctly -_-
There are probably some familiar easy steps that I'm forgetting about.
I just don't remember how to deal with these XD lol