12) Let $D$ be the domain in the Cartesian plane determined by the system of inequalities below. What is the area of $D$?
(A) $\sqrt{2} \pi$
(B) $\frac{\pi}{2}$
(C) 2
(D) $\sqrt{2}$
(E) $4-\pi$.

$\left\{\begin{array}{l}x^{2}+y^{2} \leq 1 \\ (x-1)^{2}+(y-1)^{2} \geq 1 \\ (x+1)^{2}+(y+1)^{2} \geq 1\end{array}\right.$
12) The answer is $(\mathbf{C})$.

The points of $D$ are indeed inside the circle centered at the origin with radius 1, and outside the circles centered at $(1,1)$ and $(-1,-1)$ with radius 1. In other words, $D$ consists of the dotted points in the figure alongside.

The aforementioned figure can then clearly be "cut out and reassembled" to form a square, as shown in the two figures alongside.

The area of this square is clearly 2.