could someone please tell me if i rewrote it correctly?
¬ ( ƎxƎy ¬P(x,y) ʌ ∀x∀y Q(x,y) )
to
∀xƎy P(x,y) v Ǝx∀y ¬Q(x,y)
is ¬∀x∀y the same as Ǝx∀y or ƎxƎy?
¬ ( ƎxƎy ¬P(x,y) ʌ ∀x∀y Q(x,y) )
to
∀xƎy P(x,y) v Ǝx∀y ¬Q(x,y)
is ¬∀x∀y the same as Ǝx∀y or ƎxƎy?
-
If I understand this language correctly then it should be
∀x∀y P(x,y) v ƎxƎy ¬Q(x,y)
I think of it this way. pairs of x,y have or dont have certain properties. ƎxƎy ¬P(x,y) means there is a pair (x,y) that doesnt have property P. ∀x∀y Q(x,y) means every pair (x,y) has property Q. Think of x's and y's being real numbers. Property P is that x and y are both positive, property Q is that x and y are both rational. So ( ƎxƎy ¬P(x,y) ʌ ∀x∀y Q(x,y) ) means every pair is rational (Q) and there is a pair that has at least one non-positive element. To negate that we need either every pair has both elements positive (∀x∀y P(x,y) ) OR There is a pair such that one of the elements is not rational ƎxƎy ¬Q(x,y).
∀x∀y P(x,y) v ƎxƎy ¬Q(x,y)
I think of it this way. pairs of x,y have or dont have certain properties. ƎxƎy ¬P(x,y) means there is a pair (x,y) that doesnt have property P. ∀x∀y Q(x,y) means every pair (x,y) has property Q. Think of x's and y's being real numbers. Property P is that x and y are both positive, property Q is that x and y are both rational. So ( ƎxƎy ¬P(x,y) ʌ ∀x∀y Q(x,y) ) means every pair is rational (Q) and there is a pair that has at least one non-positive element. To negate that we need either every pair has both elements positive (∀x∀y P(x,y) ) OR There is a pair such that one of the elements is not rational ƎxƎy ¬Q(x,y).