Set A has 6 elements, set B has 8, and together A and B contain 7, how many elements in A are

also in B?

Let's solve the problem step by step using set theory.

### Given:
- Number of elements in set \( A \), \( |A| = 6 \)
- Number of elements in set \( B \), \( |B| = 8 \)
- Number of elements in the union of sets \( A \) and \( B \), \( |A \cup B| = 7 \)

### Objective:
We need to find the number of elements that are in both sets \( A \) and \( B \), denoted as \( |A \cap B| \).

### Step 1: Use the principle of inclusion-exclusion.
The principle of inclusion-exclusion states:
\[
|A \cup B| = |A| + |B| - |A \cap B|
\]

### Step 2: Substitute the known values into the equation.
\[
7 = 6 + 8 - |A \cap B|
\]

### Step 3: Solve for \( |A \cap B| \).
\[
|A \cap B| = 6 + 8 - 7 = 14 - 7 = 7
\]

### Final Answer:
There are **7 elements** in set \( A \) that are also in set \( B \).

Would you like more details or have any questions?

Here are some related questions:
1. How would the scenario change if the number of elements in \( A \cup B \) were larger than either \( |A| \) or \( |B| \)?
2. Can you explain the principle of inclusion-exclusion with another example?
3. What if set \( A \) was a subset of set \( B \)?
4. How can Venn diagrams be used to solve this problem visually?
5. What if the intersection \( |A \cap B| \) was given directly—how would you calculate the union \( |A \cup B| \)?

**Tip:** Always check the logic of your calculations by considering the total number of elements in the union or intersection of sets, which should not exceed the total number in any single set.

Learn how to solve the problem involving sets A and B, where you determine the number of elements in A that are also in B using set theory and the principle of inclusion-exclusion. Detailed step-by-step solution provided.

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