To solve the problem, let's analyze it step by step. We are given:

• $X = \{-1, 1, 2\}$
• $f(x) = x^2$
• $M_1 = \{-1, -1, 1, 1, 2, 2\}$
• $M_2 = \{1, 1, -2, -2, -2\}$
• $f(M_i) = N_i$ for $i = 1, 2$

We will compute the following:

1. $f(M_1 \cup M_2)$
2. $f(M_1) \cup f(M_2)$
3. $f(M_1 \cap M_2)$
4. $f(M_1 \, \triangle \, M_2)$ (symmetric difference)
5. $f(M_1 \, \ominus \, M_2)$ (set-theoretic difference)

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Step 1: $f(x) = x^2$

Applying $f(x)$ to the elements of $M_1$ and $M_2$:

1. $f(M_1) = f(\{-1, -1, 1, 1, 2, 2\}) = \{1, 1, 1, 1, 4, 4\} = \{1, 4\}$
2. $f(M_2) = f(\{1, 1, -2, -2, -2\}) = \{1, 1, 4, 4, 4\} = \{1, 4\}$

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Step 2: Calculate $M_1 \cup M_2$, $M_1 \cap M_2$, $M_1 \triangle M_2$

1. Union ($M_1 \cup M_2$): $M_1 \cup M_2 = \{-1, -1, 1, 1, 2, 2, -2, -2\} = \{-1, 1, 2, -2\}$

2. Intersection ($M_1 \cap M_2$): Common elements in $M_1$ and $M_2 = \{1\}$.

3. Symmetric Difference ($M_1 \, \triangle \, M_2$): Elements in $M_1$ or $M_2$, but not in both = $\{-1, 2, -2\}$.

4. Set Difference ($M_1 \, \ominus \, M_2$): Elements in $M_1$ but not in $M_2 = \{-1, 2\}$.

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Step 3: Apply $f(x)$ to Each Set

1. $f(M_1 \cup M_2) = f(\{-1, 1, 2, -2\}) = \{1, 1, 4, 4\} = \{1, 4\}$
2. $f(M_1 \cap M_2) = f(\{1\}) = \{1\}$
3. $f(M_1 \triangle M_2) = f(\{-1, 2, -2\}) = \{1, 4, 4\} = \{1, 4\}$
4. $f(M_1 \ominus M_2) = f(\{-1, 2\}) = \{1, 4\}$

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Final Results:

1. $f(M_1 \cup M_2) = \{1, 4\}$
2. $f(M_1) \cup f(M_2) = \{1, 4\}$
3. $f(M_1 \cap M_2) = \{1\}$
4. $f(M_1 \triangle M_2) = \{1, 4\}$
5. $f(M_1 \ominus M_2) = \{1, 4\}$

Let me know if you would like any step explained in more detail!

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Relative Questions:

1. How do union and intersection relate to the image of a function?
2. What is the role of symmetric difference in set operations, and how does it affect the function $f$?
3. Can $f(M_1 \cup M_2)$ ever differ from $f(M_1) \cup f(M_2)$?
4. How does the choice of $f(x)$ influence the results of set operations on $M_1$ and $M_2$?
5. How would the results change if $f(x)$ were non-injective (e.g., $f(x) = \sin(x)$)?

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Tip:

Always simplify repeated elements in sets before applying functions to ensure accuracy in your computations.