To analyze whether the relation $\preceq$ defined on the set $\mathbb{R}^2$ is a partial order or an arrangement, let us examine the definition and its properties step by step:

The relation is defined as: $(x, y) \preceq (a, b) \iff x \leq a \text{ and } y \leq b.$

1. Partial Order

A partial order must satisfy the following three properties:

1. Reflexivity: $(x, y) \preceq (x, y)$ for all $(x, y) \in \mathbb{R}^2$.
2. Antisymmetry: If $(x, y) \preceq (a, b)$ and $(a, b) \preceq (x, y)$, then $(x, y) = (a, b)$.
3. Transitivity: If $(x, y) \preceq (a, b)$ and $(a, b) \preceq (c, d)$, then $(x, y) \preceq (c, d)$.

Reflexivity:

For any $(x, y) \in \mathbb{R}^2$, $x \leq x$ and $y \leq y$ are true. Hence, $(x, y) \preceq (x, y)$. Reflexivity holds.

Antisymmetry:

If $(x, y) \preceq (a, b)$, then $x \leq a$ and $y \leq b$. If $(a, b) \preceq (x, y)$, then $a \leq x$ and $b \leq y$. Combining these, $x = a$ and $y = b$. Thus, $(x, y) = (a, b)$. Antisymmetry holds.

Transitivity:

If $(x, y) \preceq (a, b)$, then $x \leq a$ and $y \leq b$. If $(a, b) \preceq (c, d)$, then $a \leq c$ and $b \leq d$. Combining these, $x \leq c$ and $y \leq d$, so $(x, y) \preceq (c, d)$. Transitivity holds.

Thus, the relation $\preceq$ is a partial order on $\mathbb{R}^2$.

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2. Arrangement

An arrangement (or total order) requires the relation to be a partial order and satisfy comparability, meaning for any two elements $(x, y), (a, b) \in \mathbb{R}^2$, either: $(x, y) \preceq (a, b) \quad \text{or} \quad (a, b) \preceq (x, y).$

In this case, comparability does not hold because it is possible to have two pairs $(x_1, y_1)$ and $(x_2, y_2)$ such that neither $(x_1, y_1) \preceq (x_2, y_2)$ nor $(x_2, y_2) \preceq (x_1, y_1)$. For example: $(1, 2) \npreceq (2, 1) \quad \text{and} \quad (2, 1) \npreceq (1, 2),$ because $1 \not\leq 2$ and $2 \not\leq 1$.

Hence, $\preceq$ is not an arrangement.

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Conclusion:

The relation $\preceq$ is a partial order on $\mathbb{R}^2$, but it is not an arrangement.

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Follow-Up Questions:

1. Can you find an example of a different partial order on $\mathbb{R}^2$?
2. How does the concept of a lattice relate to this partial order?
3. What modifications to $\preceq$ would make it a total order?
4. Can we extend this relation to higher dimensions, e.g., $\mathbb{R}^n$?
5. How does this partial order compare to lexicographical ordering on $\mathbb{R}^2$?

Tip: Always verify all three properties (reflexivity, antisymmetry, and transitivity) when evaluating partial orders.