Let us carefully compute each part of the problem, step by step.

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Step 1: Definitions and Interpretations

We are given:

• Set $X$: $X = \{-1, 1, 2\}$

• Function $f(x) = x^2$.

• Sets:

  • $M_1 = \{-1, -1, 1, 1, 2, 2\}$
  • $M_2 = \{1, 1, -2, -2, -2\}$

• $M_1 \cup M_2$: Union of the two sets.

• $M_1 \cap M_2$: Intersection of the two sets.

• $M_1 \oplus M_2$: Symmetric difference of the two sets ($M_1 \cup M_2 - M_1 \cap M_2$).

• $f(M)$: Applying $f$ to each element of $M$, resulting in a set of the images under $f$.

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Step 2: Applying the Function $f(x) = x^2$

1. Compute $f(M_1)$:

   • $M_1 = \{-1, -1, 1, 1, 2, 2\}$
   • Apply $f(x) = x^2$: $f(-1) = 1, f(1) = 1, f(2) = 4$.
   • $f(M_1) = \{1, 4\}$.

2. Compute $f(M_2)$:

   • $M_2 = \{1, 1, -2, -2, -2\}$
   • Apply $f(x) = x^2$: $f(1) = 1, f(-2) = 4$.
   • $f(M_2) = \{1, 4\}$.

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Step 3: Operations on Sets

Now compute the required expressions:

1. $f(M_1 \cup M_2)$:

   • $M_1 \cup M_2 = \{-1, 1, 2, -2\}$
   • Apply $f(x)$: $f(-1) = 1, f(1) = 1, f(2) = 4, f(-2) = 4$.
   • $f(M_1 \cup M_2) = \{1, 4\}$.

2. $f(M_1) \cup f(M_2)$:

   • $f(M_1) = \{1, 4\}, f(M_2) = \{1, 4\}$.
   • Union: $f(M_1) \cup f(M_2) = \{1, 4\}$.

3. $f(M_1 \cap M_2)$:

   • $M_1 \cap M_2 = \{1\}$
   • Apply $f(x)$: $f(1) = 1$.
   • $f(M_1 \cap M_2) = \{1\}$.

4. $f(M_1 \oplus M_2)$:

   • $M_1 \oplus M_2 = (M_1 \cup M_2) - (M_1 \cap M_2)$.
   • $M_1 \oplus M_2 = \{-1, -2, 2\}$.
   • Apply $f(x)$: $f(-1) = 1, f(-2) = 4, f(2) = 4$.
   • $f(M_1 \oplus M_2) = \{1, 4\}$.

5. $f(M_1 \ominus M_2)$:

   • This is also the symmetric difference result.
   • $f(M_1 \ominus M_2) = f(M_1 \oplus M_2) = \{1, 4\}$.

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Final Results

• $f(M_1 \cup M_2) = \{1, 4\}$
• $f(M_1) \cup f(M_2) = \{1, 4\}$
• $f(M_1 \cap M_2) = \{1\}$
• $f(M_1 \oplus M_2) = \{1, 4\}$
• $f(M_1 \ominus M_2) = \{1, 4\}$

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Would you like a detailed explanation of any step?
Here are 5 related questions to expand on the concepts:

1. How does the function $f(x)$ affect the structure of sets in general?
2. What is the difference between the union and symmetric difference of two sets?
3. Why does $f(M_1 \cap M_2)$ result in a smaller set than $f(M_1) \cup f(M_2)$?
4. How can these operations be visualized using Venn diagrams?
5. What happens when $f(x)$ is not one-to-one or onto?

Tip: Always simplify the sets first before applying a function like $f(x)$ to avoid redundant computations.