From the image, we are working with set notation and operations, particularly involving complements, intersections, and unions. Here’s an analysis of the notations provided:

Breakdown:

1. Notation Explained:

   • $A'$: The complement of set $A$, meaning all elements not in $A$. From the image, $A' = \{1, 3, 5, 7, 9\}$.
   • $C \cap D$: The intersection of sets $C$ and $D$, i.e., elements common to both $C$ and $D$.
   • $(C \cap D)'$: The complement of the intersection of $C$ and $D$, i.e., all elements not in $C \cap D$.
   • $C' \cap D'$: The intersection of the complements of $C$ and $D$.

2. Tasks:

   • Solve for $(C \cap D)'$.
   • Solve for $C' \cap D'$.

If you provide the full sets $A, C,$ and $D$, I can calculate these explicitly.

Would you like to provide more context (like the sets involved), or should I explain the general method for solving these operations?

5 Related Questions:

1. What is the general definition of a complement in set theory?
2. How does De Morgan’s Law relate $(C \cap D)'$ and $C' \cup D'$?
3. What is the result of intersecting two complements (e.g., $C' \cap D'$)?
4. How would $(C \cup D)'$ be evaluated instead?
5. Why is set complement particularly useful in probability theory?

Tip:

When working with complements and intersections, De Morgan's Laws are incredibly helpful:

• $(A \cap B)' = A' \cup B'$
• $(A \cup B)' = A' \cap B'$.