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We are given two mappings \(\Phi\) and \(\Psi\) between groups under addition. The group \( G \) and \( \overline{G} \) are both the set of real numbers \(\mathbb{R}\) under addition. 

The mappings are defined as:
- \(\Phi(x) = x\),
- \(\Psi(x) = x^2\).

We are asked to determine which of the following mappings are "operation preserving." This means the mappings should respect the group operation, i.e., addition. A mapping \( f: G \to \overline{G} \) is operation preserving if for all \( x, y \in G \), we have:
\[
f(x + y) = f(x) + f(y).
\]

### 1. Check if \(\Phi(x) = x\) is operation preserving:
For \(\Phi(x)\), we want to check if:
\[
\Phi(x + y) = \Phi(x) + \Phi(y).
\]
Since \(\Phi(x) = x\), we calculate:
\[
\Phi(x + y) = x + y \quad \text{and} \quad \Phi(x) + \Phi(y) = x + y.
\]
Both sides are equal, so \(\Phi(x)\) is operation preserving.

### 2. Check if \(\Psi(x) = x^2\) is operation preserving:
Now for \(\Psi(x)\), we check if:
\[
\Psi(x + y) = \Psi(x) + \Psi(y).
\]
We have:
\[
\Psi(x + y) = (x + y)^2 = x^2 + 2xy + y^2,
\]
while:
\[
\Psi(x) + \Psi(y) = x^2 + y^2.
\]
Since \(\Psi(x + y)\) has the additional \( 2xy \) term, we conclude that \(\Psi(x)\) is **not** operation preserving.

### Conclusion:
- \(\Phi(x) = x\) is operation preserving.
- \(\Psi(x) = x^2\) is not operation preserving.

Thus, the correct answer is:
**3) \(\Phi\) is operation preserving, and \(\Psi\) is not operation preserving.**

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Would you like further details or have any additional questions?

Here are some related questions to explore further:
1. What are other examples of operation-preserving mappings in group theory?
2. Can we generalize \(\Psi(x) = x^n\) to determine when it preserves addition?
3. How would the analysis change if the group operation was multiplication instead of addition?
4. What are homomorphisms in abstract algebra, and how do they relate to operation-preserving mappings?
5. How does the failure of operation preservation impact the structure of the group under these mappings?

**Tip:** When checking if a function is operation preserving, always compare \( f(x + y) \) with \( f(x) + f(y) \) directly by applying the function definition step by step.

This is a question on mappings between groups that are operation preserving. Let G = ℝ under addition, and Ĝ = ℝ under addition. Define the mappings Φ: G → Ĝ and Ψ: G → Ĝ by Φ(x) = x and Ψ(x) = x^2. Which of the following statements is true? 1) Φ is not operation preserving, and Ψ is not operation preserving. 2) Φ is not operation preserving, and Ψ is operation preserving. 3) Φ is operation preserving, and Ψ is not operation preserving. 4) Φ is operation preserving, and Ψ is operation preserving.

Explore operation-preserving mappings Φ and Ψ between groups of real numbers under addition. This problem involves key concepts from group theory and abstract algebra, where Φ(x) = x and Ψ(x) = x^2. Learn why Φ preserves the group operation, while Ψ does not.

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