To solve this problem, let’s analyze the given information and use it to fill in each region in the Venn diagram:

Given Information:

1. $n(A) = 25$: The total number of elements in set $A$.
2. $n(B) = 21$: The total number of elements in set $B$.
3. $n(A \cup B) = 29$: The number of elements in the union of sets $A$ and $B$.
4. $n(A') = 41$: The number of elements outside both sets $A$ and $B$.

Regions in the Diagram:

• $x$: Elements in $A$ only.
• $y$: Elements in both $A$ and $B$ (intersection).
• $z$: Elements in $B$ only.
• $w$: Elements outside both $A$ and $B$.

Solution Steps:

1. Use the Union Formula: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ Substitute the known values: $29 = 25 + 21 - n(A \cap B)$ Solving for $n(A \cap B)$: $n(A \cap B) = 25 + 21 - 29 = 17$ So, $y = 17$ (elements in both $A$ and $B$).

2. Calculate Elements Only in $A$: $x = n(A) - n(A \cap B) = 25 - 17 = 8$

3. Calculate Elements Only in $B$: $z = n(B) - n(A \cap B) = 21 - 17 = 4$

4. Calculate Elements Outside Both $A$ and $B$: Since $n(A') = 41$, we have: $w = 41$

Final Values for Each Region:

• $x = 8$
• $y = 17$
• $z = 4$
• $w = 41$

Answer Summary:

• The number of elements in region $x$ is 8.
• The number of elements in region $y$ is 17.
• The number of elements in region $z$ is 4.
• The number of elements in region $w$ is 41.

Would you like more details or have any questions?

Related Questions:

1. How would the solution change if $n(A \cup B)$ were different?
2. What is the interpretation of each region in practical terms?
3. How would adding an element common to both sets impact $n(A \cup B)$?
4. How would the regions be affected if $n(A')$ were a different value?
5. Can we solve for other unknowns if we change the value of $n(B)$?

Tip:

For Venn diagrams, always start by calculating intersections, as they help determine elements unique to each set.