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The question asks to show that the set of real numbers \( \mathbb{R} \) with respect to \( \leq \) (i.e., \( (\mathbb{R}, \leq) \)) is a partially ordered set (poset). Let's break this down.

### A **Partially Ordered Set (Poset)** is defined as:
A set \( P \) with a binary relation \( \leq \) is called a partially ordered set (or poset) if the following properties hold for all \( a, b, c \in P \):

1. **Reflexivity**: \( a \leq a \) for all \( a \in P \).
2. **Antisymmetry**: If \( a \leq b \) and \( b \leq a \), then \( a = b \).
3. **Transitivity**: If \( a \leq b \) and \( b \leq c \), then \( a \leq c \).

Now, let’s check these properties for the set \( \mathbb{R} \) (the set of real numbers) with the relation \( \leq \):

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### 1. Reflexivity:
For any real number \( a \in \mathbb{R} \), we have \( a \leq a \) (a number is always less than or equal to itself).  
This satisfies the reflexivity property.

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### 2. Antisymmetry:
If \( a \leq b \) and \( b \leq a \), then \( a = b \) for any \( a, b \in \mathbb{R} \).  
This is true for real numbers. If two real numbers are less than or equal to each other both ways, they must be equal.

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### 3. Transitivity:
If \( a \leq b \) and \( b \leq c \), then \( a \leq c \) for all \( a, b, c \in \mathbb{R} \).  
This is also true for real numbers. If one number is less than or equal to a second, and that second number is less than or equal to a third, then the first number is less than or equal to the third.

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Since all three properties—reflexivity, antisymmetry, and transitivity—hold for \( (\mathbb{R}, \leq) \), the set of real numbers \( \mathbb{R} \) with the relation \( \leq \) is a **partially ordered set (poset)**.

Would you like to explore further details or ask anything else? Here are a few related questions:

1. How can we extend the concept of a poset to lattices?
2. What is the difference between a poset and a totally ordered set?
3. Can you give examples of posets in other mathematical structures?
4. How do Hasse diagrams represent posets?
5. What are some applications of posets in computer science and data analysis?

**Tip:** When working with posets, drawing a Hasse diagram helps visualize the order relations clearly.

Show that the set of real numbers R with respect to ≤ (i.e., (R, ≤)) is a partially ordered set (poset).

Explore how the set of real numbers with the relation ≤ forms a partially ordered set (poset). This solution covers reflexivity, antisymmetry, and transitivity in detail, suitable for students in Grades 11-12.

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