Expert: Josh - 3/9/2007

Question
how i can prove this:
(Exist a X)(For All Y) P(X,Y) =>
(For All Y)(Exist a X) P(X,Y)

this is evident for me. but our teacher ask the proving!

Answer
Hi Isaac,

I thought about the symmetry property, viz. P(X,Y)=P(Y,X).
But when I look at this more carefully, it is not really saying anything. Here is why:

Translating into words, the question asks why
"There exists an X, for all values of Y, such that the proability density function P(X,Y) is......."
implies
"For all values of Y, there exists an X such that the proability density function P(X,Y) is.......(fill in blanks again)"

This is not claiming anything. It simply points out that the first two parts are "commutative". As such, no proof is necessary because this is an axiom.

Since it asserts nothing about the function P(X,Y) there is nothing to prove. It suffices to say that the order of (Exist an X) and (For all Y)  is interchangeable.