Logic and Set Theory

Help.

Can someone explain to me step by step how can I prove this statement:

∀x {(∀y p(x, y)) ⇒ q(x)} ⇔ ∀x ∃y (p(x, y) ⇒ q(x))

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en.wikipedia.org/wiki/List_of_logic_symbols
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Think about what the statements are saying. Basically, "for all y, statement p implies q if and only if there is some y for all x so that statement p implies q" the reverse being that, if there's a y for all x, then all y either work and q works (making the left statement true) or the y doesn't work, making that statement true a anyway. Thus, both statements are false when the right side fails, and then the left side fails because of that.

Thank you user you are a savior! What about direct proof?

You seem to be working in a formal system in which ∃y is taken as an axiom, as the left hand proposition makes no assertions that anything exists, and the right hand implies that if 'x's exist, then at least one y must also exist. Nothing wrong with that, but it makes me nervous.

I don't think that statement is true.

T R U T H
A
B
L
E

Brainlet here. The fuck does the upside down “A” mean?

Fuck off CS majors.

So what is the upside down “A”?

en.wikipedia.org/wiki/List_of_logic_symbols

Have fun!

Upside A is All of .. reverse E is Theres at least one ...

That proposition is false.
The predicate says for all x for all y.... if and only if for all x there exists a y....

Since y is fixed on one side and not the other, you can't prove the iff statement. Because garuanteeing there exists a fixed y says nothing about all y.

Go read any introductory chapter on first order logic and stop spamming Veeky Forums with your logic homework (I noticed your other shit threads). Use SQT

Note the brackets on the left proposition, for all x with for all y enclosed. So it would be: for all x... If and only if for all x, some y

I am aware. But for a direct proof of iff, you have to show it both directions.

So now you have to show that for all x, there exists a y implies that for all x, if for all y p (x,y) implies p (x).

based

>Can someone explain to me step by step how can I prove this statement:
You cannot, for it is false.

In order to prove it, you have to prove its parts.

The main structure is the if and only if.
So start by assuming everything on the right of the and use it to prove the lhs.

Then as a new part of the proof, assume the lhs and prove the rhs.

Going from left to right is easy. Going from right to left gives a false statement. Therefore, the iff doesn't hold. Therefore, the statement is false.

>Can someone explain to me step by step how can I prove this statement:
Yes.

perfect

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