1. For all x in real number,thetr exist a y in real number such that x<y 2. There exist a x in real number such that for all y in real number, x<y        True or False

Let's break down the two statements one by one:

1. **For all \( x \) in real numbers, there exists a \( y \) in real numbers such that \( x < y \):**

   This statement is **True**. For any real number \( x \), you can always find another real number \( y \) that is greater than \( x \). For example, if \( x = 100 \), you can choose \( y = 101 \) or \( y = 1000 \), etc. Since the real numbers are unbounded above, there is always a \( y \) such that \( x < y \).

2. **There exists an \( x \) in real numbers such that for all \( y \) in real numbers, \( x < y \):**

   This statement is **False**. There is no real number \( x \) that is less than all other real numbers because the real numbers are unbounded below. For any \( x \), you can always find a smaller \( y \). Therefore, no single \( x \) can satisfy the condition of being less than all \( y \).

### Summary:
1. **True**.
2. **False**.

Would you like further explanation or details?

Here are some related questions:
1. What is the difference between bounded and unbounded sets?
2. Can you explain the concept of real numbers being unbounded?
3. Is there a largest or smallest real number?
4. What happens in similar conditions for rational or irrational numbers?
5. How does the concept of infinity relate to real numbers?

**Tip:** Understanding how real numbers are unbounded both above and below is essential in many mathematical fields, especially calculus and analysis.

For all x in real numbers, there exists a y in real numbers such that x < y. There exists an x in real numbers such that for all y in real numbers, x < y. True or False?

This problem involves determining the truth of two statements about real numbers: (1) For all x in real numbers, there exists a y such that x < y, and (2) There exists an x such that for all y, x < y. Explore concepts of real numbers, inequalities, and unbounded sets. Suitable for Grades 10-12.

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