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This exercise describes two non-empty subsets \( A \) and \( B \) of \( \mathbb{R} \) and a scalar \( \lambda \in \mathbb{R} \). It defines some operations and then explores properties related to infima (inf) and suprema (sup) of sets.

Here are the key definitions and properties outlined:

### Definitions:
1. **Sum of Sets**: \( A + B = \{ a + b \ | \ a \in A, b \in B \} \)
2. **Product of Sets**: \( A \cdot B = \{ ab \ | \ a \in A, b \in B \} \)
3. **Scalar Multiplication**: \( \lambda A = \{ \lambda a \ | \ a \in A \} \)

### Properties (under the assumption that \( A \) and \( B \) are bounded):
- \( \inf (A + B) = \inf A + \inf B \)
- \( \sup (A + B) = \sup A + \sup B \)
  
For scalar multiplication:
- \( \inf (\lambda A) = \)
  \[
  \begin{cases}
  \lambda \inf A, & \text{if } \lambda \geq 0 \\
  \lambda \sup A, & \text{if } \lambda < 0
  \end{cases}
  \]
- \( \sup (\lambda A) = \)
  \[
  \begin{cases}
  \lambda \sup A, & \text{if } \lambda \geq 0 \\
  \lambda \inf A, & \text{if } \lambda < 0
  \end{cases}
  \]

These are classical results in the theory of sets and their bounds in real analysis, particularly when working with intervals or general bounded sets.

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Would you like me to elaborate on any of the set operations or concepts? Here are some related questions to deepen your understanding:

1. How are the infimum and supremum defined for sets in real analysis?
2. What are the implications of these properties for intervals of real numbers?
3. Can you apply these properties to specific sets \( A \) and \( B \) with concrete numbers?
4. How does scalar multiplication affect the bounds of a set depending on the sign of \( \lambda \)?
5. What is the significance of boundedness in these operations?

**Tip**: When working with sets and bounds, always check if the sets are bounded and whether the scalar involved is positive or negative, as it affects the infimum and supremum differently.

Fie A, B ⊆ ℝ, mulțimi nevide și λ ∈ ℝ. Definim mulțimile A + B = {a + b | a ∈ A, b ∈ B}, A * B = {a * b | a ∈ A, b ∈ B}, λA = {λa | a ∈ A}. Dacă A, B sunt mărginite, atunci A + B, A * B, λA sunt mărginite și avem următoarele relații pentru infim și suprem.

Explore the properties of the infimum and supremum of bounded sets in ℝ, including operations like sum, product, and scalar multiplication. Learn key real analysis concepts through well-defined set operations.

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