To find $n(A \cup B)$, which represents the number of elements in the union of sets $A$ and $B$, we will use the information provided:

• $n(A \cap B) = 50$: Number of elements in both $A$ and $B$.
• $n(A \cap B^c) = 200$: Number of elements in $A$ but not in $B$ (here, $B^c$ represents the complement of $B$).
• $n(A^c \cap B) = 150$: Number of elements in $B$ but not in $A$ (here, $A^c$ represents the complement of $A$).

Step-by-Step Solution

The total number of elements in the union of $A$ and $B$ is given by:

$n(A \cup B) = n(A \cap B) + n(A \cap B^c) + n(A^c \cap B)$

Substitute the given values:

$n(A \cup B) = 50 + 200 + 150$

Calculate the sum:

$n(A \cup B) = 400$

Conclusion

The number of elements in the union of sets $A$ and $B$ is:

$\boxed{400}$

Would you like more details or have any questions?

Related Questions

1. How would you calculate $n(A \cup B)$ if you were given $n(A)$ and $n(B)$ instead?
2. How do you determine $n(A^c \cup B^c)$ using the information given?
3. What is the relationship between $n(A \cup B)$ and $n(A^c \cap B^c)$?
4. How would $n(A \cup B)$ change if $n(A \cap B^c)$ were 300 instead of 200?
5. What formula would you use if you wanted to find $n(A \cap B^c) + n(A^c \cap B^c)$?

Tip: When working with set operations, always consider using Venn diagrams to visualize the different regions, which can help make complex problems more manageable.