homework #1 - 9: sketch the region. 1) a ∩ b ...

# Explanation: ## Step1: Recall set - notation meaning $A\cap B'$ means elements in $A$ but not in $B$. To sketch, first draw two overlapping circles representing sets $A$ and $B$. Shade the part of $A$ that does not overlap with $B$. ## Step2: For $A'\cap B'$ $A'\cap B'$ means elements that are neither in $A$ nor in $B$. Shade the area outside both $A$ and $B$ in the universal - set rectangle. ## Step3: For $A'\cup B$ $A'\cup B$ means elements that are either not in $A$ or in $B$. Shade the area outside $A$ and the entire area of $B$. ## Step4: For $A\cup B'$ $A\cup B'$ means elements that are either in $A$ or not in $B$. Shade the entire area of $A$ and the area outside $B$. ## Step5: For $A\cap B'\cup C$ First, find $A\cap B'$ (shade part of $A$ not in $B$ as before). Then, take the union with $C$, so shade the area of $C$ along with the previously shaded $A\cap B'$ area. ## Step6: For $A\cap B\cup C'$ First, find $A\cap B$ (shade the overlapping part of $A$ and $B$). Then, take the union with $C'$, so shade the area outside $C$ along with the previously shaded $A\cap B$ area. # Answer: 1. Shade the part of $A$ that does not overlap with $B$. 2. Shade the area outside both $A$ and $B$ in the universal - set rectangle. 3. Shade the area outside $A$ and the entire area of $B$. 4. Shade the entire area of $A$ and the area outside $B$. 10. Shade the part of $A$ not in $B$ and then the area of $C$. 11. Shade the overlapping part of $A$ and $B$ and then the area outside $C$.