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!