The sum of any two irrational numbers is irrational
This isnt really using quantifers but i dont know how to express on this message board.
Is irrational numbers expressed as I?
For all x, y in I,x +y is I??
There exists x and y in I such that x + y =! I?
It is customary to denote the set of rational numbers by . As far as I know, there is no equally accepted notation for the set of irrational numbers. Thus, a more standard way to denote irrational numbers is using set difference: .
Also, the following is probably understood, but just in case. If the atomic proposition (i.e., the one that cannot be broken into simpler propositions) is , then " is in " and " " are legal propositions while " is " and " " are not.
Granted, may be a well-formed formula in set theory. But when we stay at the level of the questions in the OP, it is much more likely that this formula has an error, especially given the English equivalent.
Similarly, a C compiler may produce a justifiable warning at the code "if (x = 1) then ... else ...". Even though this code is legal, it is likely to be a typo (= instead of ==).