Artificial intelligent assistant

Logic. How to deal with free variables? The task is to depict the following set: $$\left\\{ <x,y> \in \mathbb{R^2} \, |\, x^2 + y^2 > 1 \Rightarrow \exists z \, (x^2 + (y − z)^2 \leq \frac{1}{4}) \right\\}$$

The condition has the form $P(x,y) \implies Q(x,y)$, so $(x,y)$ is in the set **iff** $P(x,y)$ is false **or** $Q(x,y)$ is true.

Hence it is the union of sets of the form $\\{ (x,y) | \lnot P(x,y) \\}$ and $\\{ (x,y) | Q(x,y) \\}$.

$P(x,y)$ is false **iff** $x^2+y^2 \le 1$, hence it contains the closed unit ball.

$Q(x,y)$ is true **iff** there is some $z$ such that $(x,y) \in \overline{B}((0,z), {1 \over 2})$.

Draw a picture of the collection $\overline{B}((0,z), {1 \over 2})$, and a simpler description will become obvious.

> $\\{ (x,y) | Q(x,y) \\} = \\{ (x,y) | \ |x| \le {1 \over 2} \\}$.

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