To find the number of elements in Set $A$, let's use the formula for the union of two sets.

The formula for the union of sets $A$ and $B$ is: $|A \cup B| = |A| + |B| - |A \cap B|$ where:

• $|A \cup B|$ is the total number of elements in either Set $A$ or Set $B$,
• $|A|$ is the number of elements in Set $A$,
• $|B|$ is the number of elements in Set $B$,
• $|A \cap B|$ is the number of elements common to both sets.

Given:

• $|B| = 70$,
• $|A \cup B| = 99$,
• $|A \cap B| = 45$.

Now, substitute these values into the formula and solve for $|A|$:

$99 = |A| + 70 - 45$

Simplify the equation:

$99 = |A| + 25$

Now, isolate $|A|$:

$|A| = 99 - 25$

$|A| = 74$

Thus, Set $A$ contains 74 elements.

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Would you like more details on any of these steps or have any questions?

Here are some related questions to explore:

1. What happens if Sets $A$ and $B$ are disjoint (no elements in common)?
2. How would you calculate the intersection of three sets?
3. What if the union of Sets $A$ and $B$ had a different total than 99?
4. How does this problem change if you add another set, $C$, to the mix?
5. How can Venn diagrams help visualize problems with set intersections?

Tip: When dealing with sets, drawing a Venn diagram can often make it easier to see the relationships between sets and their elements.