The problem asks to represent the following set operations using Venn diagrams:

1. $(A \cap B)' \cap C$
2. $(A \cap B) \cup (A' \cap B')$

Let's break down each set expression before illustrating how to represent them in a Venn diagram:

1. $(A \cap B)' \cap C$

• $A \cap B$ is the intersection of sets $A$ and $B$, which means the elements common to both $A$ and $B$.
• $(A \cap B)'$ is the complement of $A \cap B$, meaning everything outside the intersection of $A$ and $B$.
• The final expression $(A \cap B)' \cap C$ represents the elements that are in $C$ but not in the intersection of $A$ and $B$.

2. $(A \cap B) \cup (A' \cap B')$

• $A \cap B$ is the intersection of $A$ and $B$.
• $A' \cap B'$ represents the intersection of the complements of $A$ and $B$, i.e., the elements that are outside both $A$ and $B$.
• The union $(A \cap B) \cup (A' \cap B')$ includes all elements either in the intersection of $A$ and $B$ or outside both $A$ and $B$.

Would you like me to generate the Venn diagrams for these expressions?