Let's walk through the problem step by step.

Problem Recap:

• Set $A$ contains 80 elements.
• Set $B$ contains 18 elements.
• Sets $A$ and $B$ have 16 elements in common: $|A \cap B| = 16$.
• There are 33 elements in $S$ (the universal set) that are not in $A$ nor in $B$.
• You need to find how many elements are in $S$ (i.e., $|S|$).

Solution:

Let’s apply the principle of inclusion-exclusion for sets.

1. Formula for union of two sets: $|A \cup B| = |A| + |B| - |A \cap B|$ Substituting the given values: $|A \cup B| = 80 + 18 - 16 = 82$

   So, there are 82 elements in either $A$, $B$, or both.

2. Elements not in $A$ or $B$: These are the 33 elements outside both sets. Therefore: |S| = |A \cup B| + \text{(elements not in A$or$B)}

3. Total number of elements in $S$: $|S| = 82 + 33 = 115$

Answer:

There are 115 elements in the universal set $S$.

Let me know if you need further clarification!
Here are 5 follow-up questions to enhance your understanding:

1. How would the formula change if three sets were involved?
2. What is the complement of a set, and how does it relate to the universal set?
3. Can two sets be disjoint? If so, what would that imply about their intersection?
4. How does the principle of inclusion-exclusion extend to more than two sets?
5. If $A \subseteq B$, what can you say about $A \cup B$ and $A \cap B$?

Tip: When working with Venn diagrams, breaking problems into smaller sections often makes them easier to solve!