1. Statement of the problem: Determine, on a two-set Venn diagram for sets $A$ and $B$ with common intersection, which regions correspond to the expressions i. $A'$, ii. $A' \cap B'$, iii. $A' \cap B$, iv. $A \cap B'$, and v. $A \cup B'$. 2. Definitions and rules used: The complement $X'$ is the set of elements in the universal set that are not in $X$. The intersection $X \cap Y$ is the set of elements common to both $X$ and $Y$. The union $X \cup Y$ is the set of elements in $X$ or in $Y$ or in both. De Morgan's laws that we will use are $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$. 3. Venn diagram regions: Label the four regions of a two-set Venn diagram for clarity. Region 1 is the part inside $A$ only, i.e. $A \setminus B$. Region 2 is the overlap $A \cap B$. Region 3 is the part inside $B$ only, i.e. $B \setminus A$. Region 4 is the part outside both $A$ and $B$, i.e. $A' \cap B'$. 4. Solutions and explanations: i. $A'$ means all points not in $A$. Therefore $A'$ consists of Region 3 and Region 4, so $A' = (B \setminus A) \cup (A' \cap B')$. Final: $A'$ corresponds to Region 3 and Region 4. ii. $A' \cap B'$ means points not in $A$ and not in $B$ simultaneously. By De Morgan this equals $(A \cup B)'$. On the diagram this is exactly Region 4 only. iii. $A' \cap B$ means points that are in $B$ but not in $A$. This is exactly Region 3 and can be written $B \setminus A$. iv. $A \cap B'$ means points that are in $A$ but not in $B$. This is exactly Region 1 and can be written $A \setminus B$. v. $A \cup B'$ means points that are in $A$ or not in $B$. That covers Region 1, Region 2, and Region 4, and excludes Region 3. An alternative identity is $A \cup B' = (B \setminus A)'$, which you can verify by De Morgan and complements. Final: $A \cup B'$ corresponds to Regions 1, 2, and 4. 5. Final concise answers: i. $A'$: Regions 3 and 4. ii. $A' \cap B'$: Region 4 only. iii. $A' \cap B$: Region 3 only. iv. $A \cap B'$: Region 1 only. v. $A \cup B'$: Regions 1, 2, and 4.