Question

Let P(x,y) be a propositional function if ꓯyꓱxP(x,y) is true does it necessarily follow that ꓱxꓯyP(x,y) is true? Justify your answer or give a counter-example

Answer 1

Answer: Ok, we know that ꓯ y ꓱ x P(x,y) is true, and suppose we are working with integers.

Lets create a counter-example

if P(x,y) : x = y, then for all integer y you can find another integer x such the proposition is true, and ꓯ y ꓱ x P(x,y) is true.

now the second part; ꓱ x ꓯyP(x,y) is true? this means that exist an x, such that p(x,y) is true for all the y in the domain. Now, is also easy to se that, for each x, there is only one y that keeps the proposition true, and is y = x.

So ꓱx ꓯy P(x,y) is not true