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The problem asks to list the irrational numbers from the given set and select all that apply. Let's first go over what irrational numbers are. An **irrational number** is a number that cannot be expressed as a fraction of two integers (i.e., it cannot be written in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers). Irrational numbers have non-repeating, non-terminating decimal expansions.

### Set provided:
\[
\left\{ -4\pi, -0.5333..., \frac{2}{13}, 0, 2, \sqrt{6}, \frac{43}{6}, 21.2, 88.777 \right\}
\]

### Analyze each element:

1. **\( -4\pi \)**: This contains \( \pi \), which is irrational. Any multiple of an irrational number is still irrational.  
   - **Irrational**.

2. **\( -0.5333... \)**: This is a decimal that repeats, so it is a rational number.  
   - **Not irrational**.

3. **\( \frac{2}{13} \)**: This is a fraction of two integers, meaning it's rational.  
   - **Not irrational**.

4. **\( 0 \)**: Zero is a rational number.  
   - **Not irrational**.

5. **\( 2 \)**: A whole number, so it is rational.  
   - **Not irrational**.

6. **\( \sqrt{6} \)**: The square root of a non-perfect square is irrational.  
   - **Irrational**.

7. **\( \frac{43}{6} \)**: A fraction of two integers, so it's rational.  
   - **Not irrational**.

8. **\( 21.2 \)**: A terminating decimal, which is rational.  
   - **Not irrational**.

9. **\( 88.777 \)**: This is a terminating decimal and thus is rational.  
   - **Not irrational**.

### Conclusion:
The irrational numbers in the set are \( -4\pi \) and \( \sqrt{6} \).

### Answering the multiple choice:
- **A**: \( \sqrt{6} \) – Irrational.
- **D**: \( -4\pi \) – Irrational.

Correct answers: **A, D**.

Would you like further explanation or details on this? 

### Related Questions:
1. What makes a number irrational?
2. Why is \( \pi \) considered irrational, and what other famous constants are irrational?
3. How can you identify irrational numbers from a set of decimals and fractions?
4. What is the difference between rational and irrational numbers?
5. Can an irrational number ever be expressed as a repeating decimal?

### Tip:
Any number involving square roots of non-perfect squares or \( \pi \) is usually irrational unless further simplified into a rational form.

List the irrational numbers from the set: {−4π, −0.5333..., 2/13, 0, 2, √6, 43/6, 21.2, 88.777}. Select all that apply.

Learn how to identify irrational numbers from a given set of numbers. This problem includes key concepts like the properties of square roots and Pi. Suitable for students in Grades 7-10.

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