What does each of the following statements mean, and which of them are true? Can you find gaming metaphors for each of these statements?
(a) For every positive number $x$, and every positive number $y$, we have $y^{2}=x$.
(b) There exists a positive number $x$ such that for every positive number $y$, we have $y^{2}=x$.
(c) There exists a positive number $x$, and there exists a positive number $y$, such that $y^{2}=x$.
(d) For every positive number $y$, there exists a positive number $x$ such that $y^{2}=x$.
(e) There exists a positive number $y$ such that for every positive number $x$, we have $y^{2}=x$.
(a) The statement means that every positive number is the square of every positive number. This is false.
In a game, an adversarial opponent picks both $x$ and $y$ from the set of positive numbers, and we have to show that $y^{2}=x$ holds no matter what the opponent chooses. We may be lucky and show that $y^{2}=x$ in some games, but we won't be able to do it for all possible choices for $x$ and $y$.
(b) The statement means that there is a positive number that is the square of every positive number. This is false.
In a game, we first pick a positive number $x$ and the adversarial opponent freely picks $y$ from the set of positive numbers. We can't pick a special $x$ such that $y^{2}=x$ no matter what the opponent chooses for $y$.
(c) The statement means some positive numbers are the square root of some others. This is true.
In a game, the adversarial opponent plays no part. We have complete control in picking positive numbers $x$ and $y$ to ensure $y^{2}=x$.
(d) The statement means that every positive number has a square. This is true.
In a game, the adversarial opponent first picks a positive number $y$. We then get to choose a number $x$ that is the square of $y$, to ensure $y^{2}=x$.
(e) The statement means there is a positive number that is the square root of every positive number. This is false.
In a game, we first choose a positive number $y$, and then an adversarial opponent picks $x$ from the set of positive numbers. We may be lucky and show that $y^{2}=x$ in some games, but we won't be able to do it for all possible choices for $x$ our opponent makes.
