The image you've provided contains text in Russian related to "Partial Orders and Equivalence Relations" in the context of set theory and algebra. I will translate and summarize the tasks for clarity.

Task Breakdown:

1. Partial Orders and Cartesian Product:

   • Verify that the inverse of any partial order on a set $X$ is a partial order on $X$.
   • Demonstrate that the Cartesian product of partial orders $R$ on $X$ and $S$ on $Y$ is a partial order on $X \times Y$.
   • Show that the union of an increasing sequence of partial orders on $X$ is a partial order on $X$.

2. Set of Partial Orders on $X$:

   • Let $\mathcal{P}(X)$ denote the set of all partial orders on $X$. Prove that the minimal elements of $\mathcal{P}(X)$ correspond precisely to linear orders on $X$.
   • Investigate whether the Cartesian product of two linear orders is again a linear order.

3. Bijections between Sets of Orders:

   • Prove there exists a bijection between the set of all partial orders on $X$ and the set of all total orders on certain rows of partial orders on $X$.
   • Similarly, show the existence of a bijection for equivalence relations on $X$ and the set of partitions of $X$.

4. Complete Linear Orders:

   • Define a complete linear order and check whether subsets of natural numbers, rationals, and reals have a supremum.

5. Subsets of $\mathcal{P}(X)$:

   • Prove that any subset of $\mathcal{P}(X)$ has a supremum and infimum with respect to the inclusion relation.

6. Factor Sets:

   • Consider equivalence relations like divisibility, parallelism, and others, and investigate how they behave as equivalence relations on sets.

Questions:

1. How can the inverse of a partial order be shown to be a partial order?
2. What are the key properties of Cartesian products when applied to partial orders?
3. How do linear orders relate to minimal elements in the set of partial orders?
4. How can bijections between partial orders and total orders be constructed?
5. What are some common examples of complete linear orders?

Tip:

When proving properties of orders, make sure to revisit the definitions of reflexivity, antisymmetry, and transitivity as these are key to showing that a relation is a partial order.

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