Let's walk through this step by step to determine the correct numbers for regions W, X, Y, and Z in the Venn diagram.

Definitions from the image:

• Universal Set (ξ): {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
• A ∩ B: {5, 10, 15} (elements in both A and B)
• B′: {7, 8, 9, 11, 12, 13, 14} (elements not in B)
• A′: {4, 6, 7, 8, 14} (elements not in A)

Interpreting the Venn Diagram Regions:

• W: Outside both A and B (only in ξ, not in A or B)
• X: In A only (A but not B)
• Y: In both A and B (intersection A ∩ B)
• Z: In B only (B but not A)

Step-by-Step Solution:

1. Intersection (Y region):

   • $A \cap B = \{5, 10, 15\}$.
   • These elements appear in both A and B, so they will go in region Y.

2. Elements only in A (X region):

   • $A' = \{4, 6, 7, 8, 14\}$ are elements not in A, but let's determine elements exclusive to A.
   • To find what belongs only to A (X region), we need to subtract the elements in $A \cap B$ from the total A.
   • So the elements in A but not in B = $\text{A} - \{5, 10, 15\}$.
   • X = $\{4, 6, 8\}$.

3. Elements only in B (Z region):

   • Elements in B only are the ones in B but not in A.
   • $B - (A \cap B) = \{7, 9, 11, 13\}$.
   • So region Z contains these elements: $\{7, 9, 11, 13\}$.

4. Elements outside both A and B (W region):

   • These are elements from ξ that do not belong to either A or B.
   • Subtract all the elements that appear in any part of the Venn diagram (A ∪ B).
   • $A \cup B = \{4, 5, 6, 7, 8, 9, 10, 11, 13, 15\}$.
   • So the elements in ξ but not in A or B are:
     $\xi - (A \cup B) = \{12, 14\}$.
   • Thus, region W = $\{12, 14\}$.

Final Answers:

• W (Outside both A and B): {12, 14}
• X (Only in A): {4, 6, 8}
• Y (In both A and B): {5, 10, 15}
• Z (Only in B): {7, 9, 11, 13}

Let me know if you have questions or need more details!

Here are five related questions:

1. How do you find the union and intersection of two sets?
2. How do complements of sets work in Venn diagrams?
3. What happens if two sets have no intersection?
4. Can an element belong to multiple regions in a Venn diagram?
5. How would the diagram change if a third set was introduced?

Tip: When working with Venn diagrams, always double-check the complements and intersections carefully to avoid missing any elements.