The uploaded image contains mathematical problems related to partial orders ("предпорядки") and equivalence relations ("эквивалентности"). I'll provide an overview of the key points in the tasks:

Problem 1

• Verify that:
  • The inversion of any partial order on a set $X$ is also a partial order on $X$.
  • The Cartesian product of two partial orders $R$ and $S$ on $X \times Y$ is a partial order on $X \times Y$.
  • The union of an increasing chain of partial orders on $X$ is still a partial order on $X$.
  • The intersection of partial orders on $X$ is also a partial order on $X$.

Problem 2

• (a) Let $\mathcal{P}(X)$ be the set of all partial orders on $X$:

  • Prove that the maximal elements in $\mathcal{P}(X) \subseteq$ correspond exactly to the linear orders on $X$.
  • Is there a minimal element in $\mathcal{P}(X) \subseteq$?

• (b) Is the Cartesian product of two linear orders also a linear order?

• (c) Prove the existence of a bijection between the set of all partial orders on $X$ and the set of all strict partial orders on $X$. Similarly, for linear orders.

• (d) Prove the existence of a bijection between the set of equivalence relations on $X$ and the set of partitions of $X$.

Problem 3

• A linear order on a set $X$ is called complete if every non-empty subset of $X$ has a supremum. Are the sets of natural numbers, rational numbers, and real numbers complete in this sense?

Problem 4

• (a) Verify that every chain of partial orders in $\mathcal{P}(X)$ has an infimum and a supremum under the inclusion relation.

• (b) Define factor sets for specific equivalence relations, such as congruence classes of integers mod 5, and parallelism of lines in a plane.

• (c) Check whether the given conditions define equivalence relations, like $x \leq y \iff f(x) = f(y)$, and discuss the equivalence in terms of certain functions.

Would you like help with any specific problem, or should I provide more detailed solutions or explanations?

Related Questions:

1. What are some properties of partial orders and linear orders?
2. How does the Cartesian product of partial orders behave compared to the Cartesian product of linear orders?
3. Can you provide examples of chains of partial orders in practice?
4. What is the significance of the bijection between partial orders and strict partial orders?
5. How does the concept of completeness differ between sets like natural numbers and real numbers?

Tip:

When working with partial orders, remember that reflexivity, antisymmetry, and transitivity are the key properties to verify in any relation to check if it's a valid partial order.