To prove that $(P, \leq)$ is a partially ordered set (POSET), we must show that the relation $\leq$, defined as "a divides b", satisfies the three properties of a partial order: reflexivity, antisymmetry, and transitivity. Let $P = \{1, 2, 3, 4, 6, 12\}$.

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1. Reflexivity:

A relation $\leq$ is reflexive if for all $a \in P$, $a \leq a$. In our case, $a \leq a$ means $a$ divides $a$, which is true for any integer since any integer divides itself.

Verification: For $a \in \{1, 2, 3, 4, 6, 12\}$, we have $a = k \cdot a$ where $k = 1$. Thus, $a \leq a$ holds for all $a \in P$.

$\therefore$ Reflexivity is satisfied.

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2. Antisymmetry:

A relation $\leq$ is antisymmetric if for all $a, b \in P$, $a \leq b$ and $b \leq a$ imply $a = b$. In our case, $a \leq b$ means $b = ka$ for some integer $k$, and $b \leq a$ means $a = m \cdot b$ for some integer $m$. If both hold, $a$ and $b$ must be the same.

Verification: Suppose $a \leq b$ and $b \leq a$. Then: $b = k \cdot a \quad \text{and} \quad a = m \cdot b.$ Substituting $b = k \cdot a$ into $a = m \cdot b$, we get: $a = m \cdot (k \cdot a) = (m \cdot k) \cdot a.$ For this to hold, $m \cdot k = 1$, which implies $m = k = 1$ (since $m$ and $k$ are integers). Thus, $a = b$.

$\therefore$ Antisymmetry is satisfied.

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3. Transitivity:

A relation $\leq$ is transitive if for all $a, b, c \in P$, $a \leq b$ and $b \leq c$ imply $a \leq c$. In our case, $a \leq b$ means $b = k \cdot a$, and $b \leq c$ means $c = m \cdot b$. We must show that $c = n \cdot a$ for some integer $n$.

Verification: Suppose $a \leq b$ and $b \leq c$. Then: $b = k \cdot a \quad \text{and} \quad c = m \cdot b.$ Substituting $b = k \cdot a$ into $c = m \cdot b$, we get: $c = m \cdot (k \cdot a) = (m \cdot k) \cdot a.$ Let $n = m \cdot k$, which is an integer. Thus, $c = n \cdot a$, meaning $a \leq c$.

$\therefore$ Transitivity is satisfied.

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Conclusion:

Since the relation $\leq$ on $P$ satisfies reflexivity, antisymmetry, and transitivity, $(P, \leq)$ is a partially ordered set (POSET).

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Would you like to explore specific examples or visualizations (e.g., a Hasse diagram) to better understand the structure?
Here are five related questions to deepen your understanding:

1. What is the difference between a partial order and a total order?
2. Can you draw the Hasse diagram for the given POSET?
3. How would adding a new element to $P$ (e.g., 8) affect the POSET structure?
4. What is the greatest lower bound (GLB) of two elements in this POSET?
5. Can a partially ordered set have cycles? Why or why not?

Tip: A Hasse diagram is a useful way to visualize a POSET by representing elements and their direct relationships (covering relations) without