Shade the Venn diagrams to determine if the sets $A \cup B'$ and $(A' \cap B)'$ are equal.

Steps to Solve

1. Shade $A \cup B'$ $B'$ is the complement of $B$, which means everything in the universal set $U$ that is not in $B$. $A \cup B'$ is the union of $A$ and $B'$, which means we shade everything that is in $A$ or in $B'$. This includes set $A$, everything outside of set $B$, and the region in $A$ that may overlap with $B$.

2. Shade $(A' \cap B)'$ First, find $A' \cap B$. $A'$ is the complement of $A$, which means everything in the universal set $U$ that is not in $A$. $A' \cap B$ is the intersection of $A'$ and $B$, which means we take the region that is in both $A'$ and $B$. This region is the part of $B$ that does not overlap with $A$. Then, $(A' \cap B)'$ is the complement of $A' \cap B$, which means everything in the universal set $U$ that is not in $A' \cap B$, so we shade everything outside of the region of $A' \cap B$. This includes set $A$, everything outside of set $B$, and the region in $A$ that may overlap with $B$.

3. Compare the shaded regions Visually compare the shaded regions in the Venn diagrams for $A \cup B'$ and $(A' \cap B)'$. If they are the same, then the sets are equal.

4. Determine if the sets are equal The shaded regions for $A \cup B'$ and $(A' \cap B)'$ are the same. Therefore, the sets are equal.