https://file.aiquickdraw.com/mathbot/file/172651332289547f1laqf.jpg

The set described in the image is:

\[
\{xy: \text{where} \, x \in \{A, B, C\} \, \text{and} \, y \in \{a, b\}^2\}
\]

This means that:
- \( x \) must be one of the letters from the set \(\{A, B, C\}\).
- \( y \) is a string formed by exactly two letters from the set \(\{a, b\}\), meaning \( y \) could be "aa", "ab", "ba", or "bb".

Now, we need to evaluate each option based on these rules:

1. **bCab**: The first character "b" is not in \(\{A, B, C\}\), so this is not valid.
2. **aBC**: The first character "a" is not in \(\{A, B, C\}\), so this is not valid.
3. **bBC**: The first character "b" is not in \(\{A, B, C\}\), so this is not valid.
4. **aCC**: The first character "a" is not in \(\{A, B, C\}\), so this is not valid.
5. **ACACA**: The string has more than 3 characters, and it doesn't follow the pattern \(xy\), where \(x\) is from \(\{A, B, C\}\) and \(y\) is a two-letter string from \(\{a, b\}^2\). So, this is not valid.
6. **Bbb**: The first character "B" is valid, and the next two characters "bb" are valid because they belong to \(\{a, b\}^2\). This is a valid string.
7. **CBa**: The first character "C" is valid, and the next two characters "ba" are valid because they belong to \(\{a, b\}^2\). This is a valid string.
8. **AAa**: The first character "A" is valid, but the next two characters "Aa" are not valid (only lowercase "a" or "b" are allowed for the last two letters). So, this is not valid.

### Valid strings:
- **Bbb**
- **CBa**

Would you like further details or have any questions?

Here are 5 related questions:
1. What would happen if \( y \) was defined as \(\{a, b\}^3\) instead of \(\{a, b\}^2\)?
2. How many total possible strings can be formed from this set?
3. If we allow both uppercase and lowercase letters for \( x \), how does the set change?
4. What if the set \( x \) included numbers in addition to letters?
5. How can we generalize this problem for any length of \( y \)?

**Tip:** Always break down the conditions of a set problem step by step to simplify string analysis.

Select all the strings that are elements of the following set: {xy: where x ∈ {A, B, C} and y ∈ {a, b}²}

Explore a set theory problem that involves selecting valid strings from a set where x ∈ {A, B, C} and y ∈ {a, b}². This problem is ideal for students in Grades 9-12, teaching the basics of set theory and string analysis.

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