To prove that $P$ is a partial order, we need to check that the relation $P$ defined on $A \times B$ satisfies the three properties of a partial order:

1. Reflexivity: $(a_1, b_1) \, P \, (a_1, b_1)$ for all $(a_1, b_1) \in A \times B$.
2. Antisymmetry: If $(a_1, b_1) \, P \, (a_2, b_2)$ and $(a_2, b_2) \, P \, (a_1, b_1)$, then $(a_1, b_1) = (a_2, b_2)$.
3. Transitivity: If $(a_1, b_1) \, P \, (a_2, b_2)$ and $(a_2, b_2) \, P \, (a_3, b_3)$, then $(a_1, b_1) \, P \, (a_3, b_3)$.

Step 1: Reflexivity

The definition of $P$ is: $(a_1, b_1) \, P \, (a_2, b_2) \iff a_1 \, P_1 \, a_2 \text{ and } b_1 \, P_2 \, b_2.$ Since $P_1$ and $P_2$ are partial orders on $A$ and $B$ respectively, they are reflexive. Thus, $a_1 \, P_1 \, a_1$ and $b_1 \, P_2 \, b_1$ for all $a_1 \in A$ and $b_1 \in B$. Hence: $(a_1, b_1) \, P \, (a_1, b_1),$ showing reflexivity of $P$.

Step 2: Antisymmetry

Suppose $(a_1, b_1) \, P \, (a_2, b_2)$ and $(a_2, b_2) \, P \, (a_1, b_1)$. By the definition of $P$, this implies: $a_1 \, P_1 \, a_2 \text{ and } b_1 \, P_2 \, b_2,$ and $a_2 \, P_1 \, a_1 \text{ and } b_2 \, P_2 \, b_1.$ Since $P_1$ and $P_2$ are partial orders, they are antisymmetric. Therefore: $a_1 = a_2 \quad \text{and} \quad b_1 = b_2.$ Hence: $(a_1, b_1) = (a_2, b_2),$ proving antisymmetry of $P$.

Step 3: Transitivity

Suppose $(a_1, b_1) \, P \, (a_2, b_2)$ and $(a_2, b_2) \, P \, (a_3, b_3)$. By the definition of $P$, this implies: $a_1 \, P_1 \, a_2 \text{ and } b_1 \, P_2 \, b_2,$ and $a_2 \, P_1 \, a_3 \text{ and } b_2 \, P_2 \, b_3.$ Since $P_1$ and $P_2$ are partial orders, they are transitive. Therefore: $a_1 \, P_1 \, a_3 \quad \text{and} \quad b_1 \, P_2 \, b_3.$ Thus: $(a_1, b_1) \, P \, (a_3, b_3),$ proving transitivity of $P$.

Conclusion

Since $P$ satisfies reflexivity, antisymmetry, and transitivity, it is a partial order on $A \times B$.

---

Do you need further details or examples? Here are five related questions you might find useful:

1. How does the direct product of partial orders differ from other types of relations?
2. Can you provide examples of $A$, $B$, and $P_1$, $P_2$ to illustrate the construction of $P$?
3. How does this concept generalize to other types of relations (e.g., equivalence relations)?
4. What is the significance of the direct product in lattice theory or set theory?
5. How would $P$ behave if $P_1$ or $P_2$ were not partial orders?

Tip: Visualizing the direct product $A \times B$ as a grid or matrix can help in understanding the behavior of the relation ( P \