Show that whether the relation (x, y) ∈ R, if, x ≥ y defined on the set of +ve integers is a partial order relation.

To show that the relation \( (x, y) \in R \), where \( x \geq y \) is defined on the set of positive integers, is a **partial order relation**, we must verify that it satisfies the three properties required for a relation to be a partial order:

1. **Reflexivity**: For all \( x \in \mathbb{Z}^{+} \) (the set of positive integers), \( x \geq x \).  
2. **Antisymmetry**: For all \( x, y \in \mathbb{Z}^{+} \), if \( x \geq y \) and \( y \geq x \), then \( x = y \).
3. **Transitivity**: For all \( x, y, z \in \mathbb{Z}^{+} \), if \( x \geq y \) and \( y \geq z \), then \( x \geq z \).

Let's check these properties step by step:

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### 1. Reflexivity:
A relation is reflexive if for every element \( x \in \mathbb{Z}^{+} \), we have \( (x, x) \in R \).

In this case, for every positive integer \( x \), we have \( x \geq x \) (since \( \geq \) is a reflexive operation). This is true for all \( x \in \mathbb{Z}^{+} \).

Therefore, the relation \( \geq \) is reflexive.

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### 2. Antisymmetry:
A relation is antisymmetric if for all \( x, y \in \mathbb{Z}^{+} \), whenever \( (x, y) \in R \) and \( (y, x) \in R \), we must have \( x = y \).

In this case, if \( x \geq y \) and \( y \geq x \), the only way both inequalities can be true simultaneously is if \( x = y \) (since both \( x \geq y \) and \( y \geq x \) imply equality).

Therefore, the relation \( \geq \) is antisymmetric.

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### 3. Transitivity:
A relation is transitive if for all \( x, y, z \in \mathbb{Z}^{+} \), whenever \( (x, y) \in R \) and \( (y, z) \in R \), it follows that \( (x, z) \in R \).

In this case, if \( x \geq y \) and \( y \geq z \), then by the properties of the inequality, we must have \( x \geq z \) (since if \( x \geq y \) and \( y \geq z \), then \( x \geq z \)).

Therefore, the relation \( \geq \) is transitive.

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### Conclusion:
Since the relation \( \geq \) on the set of positive integers satisfies **reflexivity**, **antisymmetry**, and **transitivity**, we can conclude that it is a **partial order**.

### Would you like a more detailed explanation of any of these steps, or do you have any further questions?

### Here are some related questions for further exploration:
1. What is the difference between a partial order and a total order?
2. How do we represent partial orders using Hasse diagrams?
3. Can you give an example of a set that forms a total order under the relation \( \leq \)?
4. What other relations can be considered partial orders?
5. How can we check whether a relation is an equivalence relation instead of a partial order?

**Tip:** When verifying whether a relation is a partial order, always check the three key properties: reflexivity, antisymmetry, and transitivity.

Show that whether the relation (x, y) ∈ R, if x ≥ y defined on the set of +ve integers is a partial order relation.

Discover how the relation x ≥ y, defined on the set of positive integers, is a partial order relation by verifying reflexivity, antisymmetry, and transitivity. This explanation is ideal for students in Grades 11-12 exploring advanced set theory and relations.

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