I have problem determining truth value of statements involving three quantifiers like this one.
$\forall \space x \space \exists \space y$ such that $\forall \space z, \space x+y = z,$
assuming all variables are real numbers.
I normally start these types of problems by trial and error, checking what happens if I fix one variable and vary the other. But since I have three here, I tried picking say x = 4 and z = 3 and see if I can find one y so x+y = z. Is this correct?
If so, is the statement then equivalent to the following?
$\forall \space x$ and $\forall \space z, \space \exists \space y$ such that , $\ x+y = z$
I appreciate pointers or ways to tackle this problem. Thanks.
Best Answer
The "for all x" quantifier (assuming a domain of real numbers) means that x is a real number (but no other properties are assumed). The following "there exists y" quantifier means there exists at least one real number y and that this y may depend on x. The "for all z" quantifier means that z is a real number but no other properties are assumed.
The second statement is not equivalent to the first, because the "there exists y" quantifier means that there exists at least one real number y and that this y may depend on both x and z. So the second statement is true (because y can depend on x and z, and be chosen as y = z - x) but the first statement is false because the choice of y cannot depend on z.