The image you uploaded contains two mathematical questions:

First Problem: Propositions Analysis

Problem Statement: Let $P(x, y)$ be the proposition "$x \times y$ is divisible by 5" on the set $\{2, 4, 6, 7, 9\}$. We are asked to determine which of the following propositions are true:

1. $\forall x \forall y P(x, y)$
2. $\exists x \forall y P(x, y)$
3. $\forall y \exists x P(x, y)$
4. $\exists y \exists x P(x, y)$

Solution:

• (i) $\forall x \forall y P(x, y)$ asserts that for every pair $(x, y)$ in the set, $x \times y$ is divisible by 5. This is false because $2 \times 2 = 4$ is not divisible by 5.

• (ii) $\exists x \forall y P(x, y)$ asserts there is some $x$ such that for every $y$, $x \times y$ is divisible by 5. This is false, as no single number in the set will yield products divisible by 5 with every other number.

• (iii) $\forall y \exists x P(x, y)$ asserts that for every $y$, there exists some $x$ such that $x \times y$ is divisible by 5. This is true. Consider $y=5$ (from the multiples of elements in the set), then any $x$ from the set would satisfy $x \times 5$ being divisible by 5.

• (iv) $\exists y \exists x P(x, y)$ asserts there exist $y$ and $x$ such that $x \times y$ is divisible by 5. This is clearly true, for example, $x = 5, y = 1$.

The correct answer for the first problem is E, as propositions $iii$ and $iv$ are true.

Second Problem: Logical Deduction

Given Premises:

1. If it is not raining, then the soccer game will be held.
2. If the soccer game is held, then Nam will buy a ticket.
3. Nam did not buy a ticket.

Question: Which conclusion can be drawn?

Solution: From the premises:

• Nam did not buy a ticket implies the soccer game was not held.
• If the soccer game was not held, it must have been raining (as the game would have been held if it were not raining).

The correct conclusion is B. It did not rain.

Translation to Vietnamese:

Câu hỏi đầu tiên: Phân tích mệnh đề Đặt $P(x, y)$ là mệnh đề "$x \times y$ chia hết cho 5" trên tập hợp $\{2, 4, 6, 7, 9\}$. Chúng ta được yêu cầu xác định các mệnh đề nào sau đây là đúng:

1. $\forall x \forall y P(x, y)$
2. $\exists x \forall y P(x, y)$
3. $\forall y \exists x P(x, y)$
4. $\exists y \exists x P(x, y)$

Câu trả lời cho câu hỏi đầu tiên là E, vì mệnh đề $iii$ và $iv$ là đúng.

Câu hỏi thứ hai: Suy luận logic Giả thiết đã cho:

1. Nếu trời không mưa, trận bóng đá sẽ được tổ chức.
2. Nếu trận bóng đá được tổ chức, thì Nam sẽ mua vé.
3. Nam đã không mua vé.

Câu hỏi: Kết luận nào có thể được rút ra?

Giải pháp: Từ các giả thiết:

• Nam không mua vé có nghĩa là trận bóng đá không được tổ chức.
• Nếu trận bóng đá không được tổ chức, phải là trời đã mưa (vì trận đấu sẽ được tổ chức nếu trời không mưa).

Kết luận chính xác là B. Trời đã mưa.

Would you like more details or have any questions? Here are a few related questions to consider:

1. Why can't a single number in the set satisfy $x \times y$ being divisible by 5 for all $y$?
2. How do you determine all possible pairs $(x, y)$ such that $x \times y$ is divisible by 5?
3. Can you explore more about logical implications and how they are used in problem solving?
4. What would be an example of a case where the first premise is directly tested (related to it raining or not)?
5. Could there be a situation in logic where Nam not buying a ticket does not necessarily mean it rained?

Tip: When solving logic puzzles or propositions, it can help to write down all possible outcomes or scenarios based on the given statements to visually see the relationships and constraints.