Let's go through the image step by step and solve the problems.

Part A: Venn Diagram

1. $U$ (Universal Set)

The universal set $U$ consists of all elements inside and outside the circles, which in this case are the letters: $U = \{a, b, c, d, e, f, g, h, i, j\}$

2. $A' \cup B'$

• $A'$ is the complement of set $A$, meaning all elements not in $A$. The elements outside $A$ are $\{d, e, f, g, h, i, j\}$.
• $B'$ is the complement of set $B$, meaning all elements not in $B$. The elements outside $B$ are $\{a, b, h, i, j\}$.

The union $A' \cup B'$ consists of all elements in $A'$ or $B'$: $A' \cup B' = \{a, b, d, e, f, g, h, i, j\}$

3. $A' \cap B'$

The intersection $A' \cap B'$ consists of elements that are both in $A'$ and $B'$. This would be the elements outside both $A$ and $B$, which are: $A' \cap B' = \{h, i, j\}$

4. $A' \cap B$

This is the intersection of the complement of $A$ and set $B$. So, we are looking for elements in $B$ that are not in $A$: $A' \cap B = \{d, f\}$

5. $A \cup B'$

This is the union of set $A$ and the complement of $B$. The elements in $A$ are $\{a, b, c\}$ and the elements in $B'$ are $\{a, b, h, i, j\}$. The union is: $A \cup B' = \{a, b, c, h, i, j\}$

Part B: Given sets $A = \{2, 4, 5, 8\}$, $B = \{3, 4, 6, 8\}$, and $U = \{1, 2, 3, 4, 5, 6, 7, 8\}$

1. $A'$

$A'$ is the complement of set $A$, meaning all elements in the universal set $U$ that are not in $A$. So: $A' = \{1, 3, 6, 7\}$

Would you like further clarification on any of these solutions? Let me know if you'd like more details or if you have any other questions.

Here are 5 related questions you can consider:

1. What is the intersection of $A \cap B$ from Part B?
2. What is $B' \cup A'$ using the sets from Part B?
3. Can you visualize how the complement of a set affects its union or intersection?
4. How would you solve for $A \cap B'$ in Part B?
5. How can you verify your Venn diagram answers by comparing them with the actual sets?

Tip: In Venn diagrams, intersections represent the common elements between sets, while unions gather elements from both sets.