We are given the following information:

• $n(A \cup B)' = 5$, which means the number of elements in the complement of $A \cup B$ is 5.
• $n(A - B) = 3$, which means the number of elements in $A$ but not in $B$ is 3.
• $n(B) = 7$, which means the number of elements in $B$ is 7.

We need to find $n(U)$, the number of elements in the universal set $U$.

Step-by-step solution:

1. From the complement, we know: $n(A \cup B)' = n(U) - n(A \cup B)$ So, $5 = n(U) - n(A \cup B)$ This gives: $n(A \cup B) = n(U) - 5$

2. Next, use the formula for the union of two sets: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$

3. We know $n(A - B) = 3$, which represents the part of $A$ not in $B$. Therefore, the total number of elements in $A$ can be written as: $n(A) = n(A - B) + n(A \cap B)$ So, $n(A) = 3 + n(A \cap B)$

4. Substitute this into the formula for $n(A \cup B)$: $n(A \cup B) = (3 + n(A \cap B)) + 7 - n(A \cap B)$ Simplifying: $n(A \cup B) = 3 + 7 = 10$

5. Now, we know that $n(A \cup B) = n(U) - 5$, so: $10 = n(U) - 5$ Solving for $n(U)$: $n(U) = 15$

Thus, the number of elements in the universal set $n(U) = 15$.

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Would you like more details or have any questions about the solution?

Here are 5 related questions to deepen your understanding:

1. How do you calculate $n(A \cap B)$ if more data was provided?
2. What is the role of complements in set theory problems like this?
3. How can you extend this problem if $n(A \cup B)$ were directly given?
4. How does $n(A - B)$ differ from $n(A \cup B)$ in interpretation?
5. How would the answer change if the value of $n(B)$ was different?

Tip: When solving set problems, always sketch a Venn diagram—it helps visualize the relationships between the sets.