I have a logic based statement of this form:
$$ (\forall x\ ,p(x))\to (\exists y\ , q(x,y))) $$
I am trying to find out if this statement is TRUE or False. I have 2 methods of proof, and one leads to FALSE, the other leads to TRUE. I have a problem hence!
MY METHOD PART 1:
So for the first part of this logic equation I call it P and the second part I call it Q, so i get an overall implication equation:
$$ P \to Q $$
So I thought I could do a counter-example pick an x-value in order to set P to be TRUE and then pick a y-value to make Q-false. That way in an implication statement if P=TRUE and you have Q=FALSE, then overall the implication is FALSE which would disprove the overall statement.
I have seen on some other site that you can do this in order to disprove this.
BUT, I now have doubts.
MY METHOD PART 2:
Because in the first part it has the "For All" quantifier which in my notation is inside P. SO now i will take the "For ALL" into account. . I can show that the function inside P, p(x), I can find a counter-example x-value to show that the left side, capital P as FALSE.
NOW in an IMPLICATION if the first part P is FALSE, then based on truth-table for this, it does not matter if Q is TRUE or FALSE. Because as soon as P is False it makes the whole implication statement TRUE!!!
SO now I did 2 Methods, the first one shows that the overall statement is "FALSE" and I have a second Method that shows that the overall statement is "TRUE".
SO I am not sure which method is the problem.
Hope someone can clarify this.