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.