Question

(a) shade ((a cap b)).
(b) shade (a cup b).

Explanation:

Step1: Simplify (a) using De Morgan's Law

Recall De Morgan's Law: $(X \cap Y)' = X' \cup Y'$. Let $X=A$, $Y=B'$, so:
$$(A \cap B')' = A' \cup (B')' = A' \cup B$$
This means the region is all elements not in $A$, plus all elements in $B$. Visually, this is the entire set $B$ and the part of the universal set outside $A$.

Step2: Identify region for (a)

Shade:

1. The entire circle $B$ (including its overlap with $A$)
2. The area inside the universal set $U$ that is outside circle $A$

Step3: Simplify (b) using set definitions

$A \cup B'$ is all elements in $A$, plus all elements not in $B$. Visually, this is:

1. The entire circle $A$ (including its overlap with $B$)
2. The area inside the universal set $U$ that is outside circle $B$

Answer:

(a) Shade the entire circle $B$ and the area of $U$ outside circle $A$.
(b) Shade the entire circle $A$ and the area of $U$ outside circle $B$.

Visual summary (text description):

• For $(A \cap B')'$: The shaded region covers all of $B$ and the non-$A$ part of the universal set.
• For $A \cup B'$: The shaded region covers all of $A$ and the non-$B$ part of the universal set.