Let's first address the question on the uploaded image:

1. Task:

You are asked to list the elements of $(X' \cup Y' \cup Z')'$, which can be interpreted as finding the complement of the union of the complements of sets $X$, $Y$, and $Z$.

Step-by-step:

• $X'$ is the complement of set $X$, i.e., elements that are not in $X$. From the diagram, the elements outside $X$ are: $X' = \{f, h, j, k, l, u, v, w, s, e\}.$

• $Y'$ is the complement of set $Y$, i.e., elements that are not in $Y$. From the diagram, the elements outside $Y$ are: $Y' = \{b, r, i, d, g, f, j, k, l, u, v, w\}.$

• $Z'$ is the complement of set $Z$, i.e., elements that are not in $Z$. From the diagram, the elements outside $Z$ are: $Z' = \{c, b, r, i, d, g, a, x, m, e, s\}.$

Now we take the union of these complements $X' \cup Y' \cup Z'$. This set includes all the unique elements from the complements: $X' \cup Y' \cup Z' = \{f, h, j, k, l, u, v, w, s, e, b, r, i, d, g, a, x, m\}.$

Finally, the complement of this union, $(X' \cup Y' \cup Z')'$, includes the elements that are in none of the complements, i.e., the elements inside all sets $X$, $Y$, and $Z$: $(X' \cup Y' \cup Z')' = \{c\}.$

2. Task:

The next section provides sets based on mathematical criteria:

• $x$ is an integer.
• $A = \{ x : 41 \leq x \leq 50 \}$, so the set $A$ is $\{41, 42, 43, 44, 45, 46, 47, 48, 49, 50\}$.
• $B = \{ x : x \text{ is a multiple of } 3 \}$, which will include integers that are divisible by 3.
• $C = \{ x : x \text{ is a prime number} \}$, a prime number is a number greater than 1 with no divisors other than 1 and itself.

You are asked to complete the Venn diagram for these sets.

Would you like more details on these steps or have any specific questions?

Here are some related questions for better understanding:

1. How do we find the complement of a set?
2. What is the difference between the union and intersection of sets?
3. How can Venn diagrams help in understanding set relationships?
4. How are prime numbers distributed within a given range?
5. Can an element belong to multiple sets simultaneously?

Tip:

When solving set problems involving complements and unions, always start by identifying elements in each set, then move on to apply the operations in sequence to avoid confusion.