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.