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.

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 $n$.