So, would you say this like:
"For the set of all real numbers x, if x does not equal 5 then their exists (at least) one y in the set of real numbers such that y is less than x."
A is "For the set of all real numbers x, x not equal to 5."
B is "There exists one y in the set of real numbers such that y is less than x"
A B B A.
The negation is A B
And you would say, "For the set of all real numbers x, x not equal to 5 and the set of all real numbers y, y is greater than x."
So, can you simplify this to, "for the set of all real numbers, y is greater than x, with the exception of x equals 5, where x is not defined"?
Yes, I just found this link, which might be helpful: An Elementary Introduction to Logic and Set Theory: Methods of Proof
And there's a section Rules of Replacement for Quantified Predicates with an example, which shows that what I wrote in my last post is definately wrong, but now, I wonder what the negation of (x) is?!
does that even make sense?
Usually we have a propositional function.
Example: Let P(x) mean that x is a rational number.
Now reads “x is not a rational number.”
reads: “It is false that every x is rational.
reads: “Some x is not rational.
Clearly those two statements are equivalent.