2. (a) Define our "universe" S as the region in the xy-plane that is described by
$S = \{(x, y) : |x| + 2|y| \le 2 \text{ and } x < \frac{1}{2} \}$.
There are an infinite number of solutions to this problem, so I expect to see as many different
solutions as there are students in the class.
i. Sketch S in the xy-plane.
ii. Mathematically and graphically describe three sets $A_1$, $A_2$, and $A_3$ taken from S that are
mutually exclusive but not collectively exhaustive.
iii. Mathematically and graphically describe three sets $B_1$, $B_2$, and $B_3$ taken from S that are
collectively exhaustive but not mutually exclusive.
(b) Let A = $\{(x, y) : 2x \ge -y\}$, B = $\{(x, y) : x \ge 2y\}$, and C = $\{(x, y) : x^2 + y^2 < 9\}$. Being as
accurate as possible, sketch $(A \cup B^c \cup C) \cap (A^c \cup B \cup C)$.
