Let A be the set of real numbers R. Express the following proposition with a simple sentence. ∀ x ∈ A ∃ y ∈ A [(x < y) ∧ (∀ z ∈ A (x < z) ⇒ (y ≤ z))] Is this proposition TRUE ? If not, is there a set A for which it is TRUE ? Justify your answer.

### Proposition:
The proposition states:  
**For every real number \( x \), there exists a real number \( y \) such that \( x < y \) and for all real numbers \( z \), if \( x < z \), then \( y \leq z \).**

### Interpretation:
In simpler terms, this proposition says that for any real number \( x \), there is a number \( y \) that is greater than \( x \), and \( y \) is the smallest number greater than \( x \) in the set \( A \).

### Answer:
This proposition is **false** if \( A = \mathbb{R} \) (the set of all real numbers).

#### Reason:
In the real numbers, there is no smallest number greater than any given real number \( x \). This is because between any two real numbers, there is always another real number. In other words, the real numbers are dense, and there is no "next" real number after any \( x \). Therefore, there is no single \( y \) that satisfies the condition of being the smallest number greater than \( x \).

### Is there a set \( A \) for which this proposition is **true**?
Yes, the proposition would be true if \( A \) is the set of integers \( \mathbb{Z} \). In the set of integers, for every integer \( x \), there is a smallest integer \( y \) such that \( x < y \) (which would simply be \( y = x + 1 \)). The integers do not have the same density property as the real numbers, so a smallest integer greater than \( x \) exists.

### Conclusion:
- The proposition is **false** for \( A = \mathbb{R} \).
- The proposition is **true** for \( A = \mathbb{Z} \) (the set of integers).

### Would you like any further details or clarifications? 

Here are 5 related questions to deepen your understanding:
1. How does the density property of real numbers affect the existence of a smallest number greater than \( x \)?
2. How does this proposition relate to concepts like infimum and supremum in set theory?
3. What are some other sets where the proposition might hold true?
4. How would the proposition change if we asked for a maximum \( y \) instead of a minimum?
5. What are other examples of sets where a smallest number greater than \( x \) does exist?

**Tip:** When working with sets and propositions, always consider the properties of the set, such as density, continuity, or discreteness. These can drastically affect the truth of mathematical statements.

Let A be the set of real numbers R. Express the following proposition with a simple sentence. ∀ x ∈ A ∃ y ∈ A [(x < y) ∧ (∀ z ∈ A (x < z) ⇒ (y ≤ z))] Is this proposition TRUE? If not, is there a set A for which it is TRUE? Justify your answer.

Explore a complex proposition involving the set of real numbers (R) and learn whether it is true. Understand the mathematical logic behind quantifiers and the density property of real numbers. Suitable for undergraduate students studying advanced set theory.

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