Let the Universal Set, S, have 162 elements. A and B are subsets of S. Set A contains 94 elements and Set B contains 35 elements. If the total number of elements in either A or B is 123, how many elements are in B but not in A?


To solve this problem, we can use the principle of inclusion-exclusion for sets. Here’s how the formula works:

\[
|A \cup B| = |A| + |B| - |A \cap B|
\]

Where:
- \( |A \cup B| \) is the total number of elements in either set \( A \) or set \( B \), given as 123.
- \( |A| \) is the number of elements in set \( A \), given as 94.
- \( |B| \) is the number of elements in set \( B \), given as 35.
- \( |A \cap B| \) is the number of elements common to both sets \( A \) and \( B \), which we need to find.

### Step 1: Find \( |A \cap B| \)
Plug the given values into the inclusion-exclusion formula:

\[
123 = 94 + 35 - |A \cap B|
\]

\[
123 = 129 - |A \cap B|
\]

Solving for \( |A \cap B| \):

\[
|A \cap B| = 129 - 123 = 6
\]

Thus, there are 6 elements common to both sets \( A \) and \( B \).

### Step 2: Find the number of elements in \( B \) but not in \( A \)
The number of elements in \( B \) but not in \( A \) is given by:

\[
|B - A| = |B| - |A \cap B|
\]

\[
|B - A| = 35 - 6 = 29
\]

Therefore, the number of elements in \( B \) but not in \( A \) is 29.

Would you like more details or have any questions?

Here are five related questions to further explore this topic:
1. How would the solution change if the total number of elements in either \( A \) or \( B \) were different?
2. Can you determine the number of elements in \( A \) but not in \( B \)?
3. What would happen if \( |A| \) or \( |B| \) exceeded the total number of elements in \( S \)?
4. How can you represent this problem using a Venn diagram?
5. How does the principle of inclusion-exclusion extend to three sets?

**Tip:** Always double-check your calculations when using formulas to ensure no steps are missed or misinterpreted.

Let the Universal Set, S, have 162 elements. A and B are subsets of S. Set A contains 94 elements and Set B contains 35 elements. If the total number of elements in either A or B is 123, how many elements are in B but not in A?

Learn how to find the number of elements in Set B but not in Set A using the Inclusion-Exclusion Principle. This problem involves a universal set with 162 elements, with subsets A and B containing 94 and 35 elements respectively. A detailed step-by-step solution is provided. Suitable for students in Grades 9-11.

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