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The image contains two mathematical problems in Indonesian related to set theory. Here's a transcription and solution for each:

### 3. Misalkan \( X = \{1, 2\} \), \( Y = \{a\} \), dan \( Z = \{\alpha, \beta\} \). Daftarkan anggota-anggota dari masing-masing himpunan:
a. \(- X \times Y \times Z\)  
b. \( X \times X \times X \times X \times X\)

#### a. \(- X \times Y \times Z\)
This problem involves the Cartesian product of the sets \( X \), \( Y \), and \( Z \). The Cartesian product \( X \times Y \times Z \) contains all ordered triples formed by taking one element from each set.

- \( X = \{1, 2\} \)
- \( Y = \{a\} \)
- \( Z = \{\alpha, \beta\} \)

The Cartesian product is:

\[
X \times Y \times Z = \{(1, a, \alpha), (1, a, \beta), (2, a, \alpha), (2, a, \beta)\}
\]

#### b. \( X \times X \times X \times X \times X \)

This is the Cartesian product of \( X \) with itself five times, forming ordered 5-tuples:

- \( X = \{1, 2\} \)

Each element of \( X \times X \times X \times X \times X \) is a 5-tuple where each component is either 1 or 2. There are \( 2^5 = 32 \) such tuples, and they are:

\[
\{(1, 1, 1, 1, 1), (1, 1, 1, 1, 2), \dots, (2, 2, 2, 2, 1), (2, 2, 2, 2, 2)\}
\]

### 4. Tentukan himpunan kuasa dari himpunan \( \{\emptyset, \{\emptyset\}\} \) dan tentukan berapa banyak elemen pada himpunan \( P (\{\emptyset, a, \{a\}, \{\{a\}\}\}) \)?

#### a. Power Set of \( \{\emptyset, \{\emptyset\}\} \)

The power set of a set contains all its subsets, including the empty set and the set itself.

For \( A = \{\emptyset, \{\emptyset\}\} \), the subsets are:

\[
\mathcal{P}(A) = \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset, \{\emptyset\}\}\}
\]

So, the power set has 4 elements.

#### b. \( P (\{\emptyset, a, \{a\}, \{\{a\}\}\}) \)

We need to find the number of elements in the power set of \( B = \{\emptyset, a, \{a\}, \{\{a\}\}\} \). The number of elements in a power set is given by \( 2^n \), where \( n \) is the number of elements in the set.

Here, \( B \) has 4 elements, so:

\[
|\mathcal{P}(B)| = 2^4 = 16
\]

Thus, the power set has 16 elements.

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Would you like more details or have any questions? Here are 5 related questions:

1. How is the Cartesian product defined for two or more sets?
2. What is the significance of the empty set \( \emptyset \) in set theory?
3. How do you determine the number of elements in a Cartesian product?
4. What is the power set, and how is it calculated for a given set?
5. How do subsets relate to the original set in set theory?

**Tip**: When calculating power sets, remember that each element of a set either belongs or doesn’t belong to a subset, which leads to \( 2^n \) possibilities for a set of size \( n \).

3. Misalkan X = {1, 2}, Y = {a} dan Z = {α, β}. Daftarkan anggota-anggota dari masing-masing himpunan:
a. - X × Y × Z
b. X × X × X × X × X

4. Tentukan himpunan kuasa dari himpunan {∅, {∅}} dan tentukan berapa banyak elemen pada himpunan P({∅, a, {a}, {{a}}})?

Solve set theory problems including Cartesian product and power set. Learn how to list members of sets and calculate the number of elements in power sets. Perfect for students in Grades 9-11.

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