1. Let X = {a, b, c}, Y = {a, b, c}, and Z = {a, b, b, c, c, c }.

a. What are the elements of X, Y, and Z? How are X, Y, and Z related? 

b. How many elements are in the set { a, {a, b}, {a} } ?

c. For each positive integer x, let Ax = {x, x2}. Find A1, A2, and A3.

Let's break down the problem into its parts:

### Part (a): Elements of \( X \), \( Y \), and \( Z \)
- \( X = \{a, b, c\} \)
- \( Y = \{a, b, c\} \)
- \( Z = \{a, b, b, c, c, c\} \)

#### How are \( X \), \( Y \), and \( Z \) related?
- \( X \) and \( Y \) are **sets** that contain the exact same elements: \( \{a, b, c\} \). Since sets do not allow duplicates, both \( X \) and \( Y \) contain **three unique elements**: \( a \), \( b \), and \( c \).
- \( Z \), however, contains repetitions of the elements \( b \) and \( c \). If we treat \( Z \) as a **multiset**, it has the following structure:
  - Element \( a \) occurs **once**.
  - Element \( b \) occurs **twice**.
  - Element \( c \) occurs **three times**.

Therefore, \( X \), \( Y \), and \( Z \) are related in that they contain the same elements, but \( Z \) allows for repetition, unlike \( X \) and \( Y \), which are traditional sets.

### Part (b): Number of Elements in the Set \( \{ a, \{a, b\}, \{a\} \} \)
This set has the elements:
1. \( a \) (an element itself),
2. \( \{a, b\} \) (a set containing two elements),
3. \( \{a\} \) (a set containing the single element \( a \)).

Even though one of the elements is a set, the question is asking for the number of elements in the overall set, so the set \( \{ a, \{a, b\}, \{a\} \} \) contains **3 elements**.

### Part (c): Finding \( A_1 \), \( A_2 \), and \( A_3 \)
For each positive integer \( x \), \( A_x = \{x, x^2\} \). Let's find these values for \( x = 1, 2, 3 \):

- \( A_1 = \{1, 1^2\} = \{1, 1\} \), which simplifies to \( A_1 = \{1\} \).
- \( A_2 = \{2, 2^2\} = \{2, 4\} \).
- \( A_3 = \{3, 3^2\} = \{3, 9\} \).

### Recap:
- (a) \( X = \{a, b, c\}, Y = \{a, b, c\}, Z = \{a, b, b, c, c, c\} \). \( Z \) is a multiset, whereas \( X \) and \( Y \) are sets.
- (b) The set \( \{ a, \{a, b\}, \{a\} \} \) has **3 elements**.
- (c) \( A_1 = \{1\}, A_2 = \{2, 4\}, A_3 = \{3, 9\} \).

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Would you like more details or have any other questions?

Here are 5 related questions:
1. How does a set differ from a multiset?
2. Can a set contain other sets as elements?
3. What is the cardinality of a power set of \( X = \{a, b, c\} \)?
4. If \( Z \) were a proper set instead of a multiset, what would it look like?
5. How are ordered pairs different from sets in set theory?

**Tip:** Remember, sets do not allow duplicates, but multisets do. Always check how the collection is defined!

1. Let X = {a, b, c}, Y = {a, b, c}, and Z = {a, b, b, c, c, c }.

a. What are the elements of X, Y, and Z? How are X, Y, and Z related?
b. How many elements are in the set { a, {a, b}, {a} }?
c. For each positive integer x, let Ax = {x, x^2}. Find A1, A2, and A3.

Explore set theory concepts such as the elements of X, Y, and Z, how multisets differ from sets, and how to calculate the number of elements in a set with nested sets. This problem also introduces the calculation of A1, A2, and A3 for quadratic sets. Suitable for Grades 9-12.

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