https://r2.erweima.ai/mathbot/image/1727070978235krr6fnlc.jpg

The image contains a series of mathematical expressions written in Persian. The task is to identify whether each expression is true or false. The expressions seem to involve set theory, inequalities, and real numbers. Let’s break down each statement.

### 1. \( 4 \in \left( -\frac{7}{2}, -1 \right] \)
   - Interpretation: The number 4 is claimed to be an element of the interval \((-3.5, -1]\).
   - Since 4 is not within this interval, **this statement is false**.

### 2. \( -2 \in (-2, 0) \)
   - Interpretation: The number -2 is claimed to be an element of the interval \((-2, 0)\), which is an open interval (does not include -2).
   - **This statement is false** because the interval does not include -2.

### 3. \( e = (-\pi, 0) \)
   - Interpretation: This seems to claim that the constant \(e\) (approximately 2.718) is equal to the interval \((- \pi, 0)\).
   - **This statement is false** because a number cannot equal an interval.

### 4. \( \{-7, 1\} \subseteq [-7, 1] \)
   - Interpretation: The set \(\{-7, 1\}\) is claimed to be a subset of the interval \([-7, 1]\), which is a closed interval.
   - **This statement is true** because both -7 and 1 are within the closed interval \([-7, 1]\).

### 5. \( 0 \in [-\pi, 2) \)
   - Interpretation: The number 0 is claimed to be an element of the interval \([-\pi, 2)\).
   - **This statement is true** because 0 is indeed in the interval \([-\pi, 2)\).

### 6. \( \sqrt{2} \in (0, 1) \)
   - Interpretation: The number \(\sqrt{2}\) (approximately 1.414) is claimed to be in the interval \((0, 1)\).
   - **This statement is false** because \(\sqrt{2}\) is greater than 1.

### Summary of Answers:
1. False
2. False
3. False
4. True
5. True
6. False

Would you like to dive into more details on any of these? 

Here are five related questions:
1. What is the difference between open and closed intervals?
2. How can you check if a number belongs to a given interval?
3. What is the significance of the constant \( e \)?
4. How do you determine if one set is a subset of another?
5. How does the square root of a number relate to inequalities?

**Tip:** Always be mindful of whether an interval is open or closed when checking if a number belongs to it.

Determine the truth value of each mathematical statement involving sets, intervals, and real numbers.

Learn how to determine the truth value of mathematical statements involving sets, intervals, and real numbers. This problem involves core concepts in set theory, interval notation, and real analysis. Suitable for students in Grades 9-12.

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