Let's analyze each of the statements given in the image and determine if they are true or false:

1. Statement 1: "Пусть $X$ — множество людей на планете, $m, n \in X$, тогда отношение $m \sim n$ тогда и только тогда, когда $m$ и $n$ знакомы. Это отношение является отношением эквивалентности."

   • Here, $m \sim n$ means $m$ and $n$ are "znakom" (which typically implies some form of acquaintance or familiarity). To be an equivalence relation, the relation must satisfy three properties: reflexivity, symmetry, and transitivity. Acquaintance is usually symmetric and reflexive, but may not be transitive, as someone known by one person might not be known by another in a network. Therefore, this is not necessarily an equivalence relation.

2. Statement 2: "Отношение сравнимости по модулю $n$ является отношением эквивалентности."

   • Modulo $n$ congruence is indeed an equivalence relation on the integers. It satisfies reflexivity (any integer $a \equiv a \mod n$), symmetry (if $a \equiv b \mod n$, then $b \equiv a \mod n$), and transitivity (if $a \equiv b \mod n$ and $b \equiv c \mod n$, then $a \equiv c \mod n$). So, this statement is true.

3. Statement 3: "Отношение параллельности двух прямых является отношением эквивалентности."

   • The relationship of parallelism between lines is an equivalence relation because it is reflexive (a line is parallel to itself), symmetric (if line $l_1$ is parallel to line $l_2$, then $l_2$ is parallel to $l_1$), and transitive (if $l_1$ is parallel to $l_2$ and $l_2$ is parallel to $l_3$, then $l_1$ is parallel to $l_3$). Therefore, this statement is true.

4. Statement 4: "Отношение порядка является отношением эквивалентности на множестве натуральных чисел."

   • The ordering relation (like $\leq$) on natural numbers is not an equivalence relation because it is not symmetric. For example, if $a \leq b$, it does not imply that $b \leq a$ unless $a = b$. Thus, this statement is false.

Summary

The true statements are:

• Statement 2: The congruence relation modulo $n$ is an equivalence relation.
• Statement 3: The relation of parallelism between lines is an equivalence relation.

Would you like more details on any of these points or have any further questions?

Here are five related questions for further exploration:

1. What are the main properties of an equivalence relation?
2. How is modular arithmetic used in number theory and cryptography?
3. Can you provide an example where the acquaintance relation is not transitive?
4. Why does order not qualify as an equivalence relation on natural numbers?
5. Are there other geometric relations, besides parallelism, that are equivalence relations?

Tip: When determining if a relation is an equivalence relation, always check if it satisfies reflexivity, symmetry, and transitivity.