https://file.aiquickdraw.com/mathbot/file/1726336124969mwf6jogh.jpg

The exercise shown in the image is about set theory, specifically dealing with *membership* (∈) and *subset* (⊂) relations. The goal is to fill the blank spaces with either the symbol "∈" (belongs to) or "⊂" (is a subset of) to make the propositions true.

Let's go through the exercise step by step:

1. **{2, 6, 4} ? {2, 2, 6, −6, −3}**
   - Here, the set on the left is **{2, 6, 4}**. We need to determine whether it is a subset or an element of the set on the right, **{2, 2, 6, −6, −3}**.
   - Since **all the elements of {2, 6, 4} are not present** in {2, 2, 6, −6, −3} (note that 4 is missing), **{2, 6, 4}** is **not a subset** of the right set, so it can't be "⊂".
   - Also, **{2, 6, 4}** is a **set itself**, not an element of the right set, so the answer is neither "∈."
   - Therefore, this proposition is false under both conditions, but for the exercise, it's better to use "⊂" (since we can't insert something like 'false').

2. **ℤ ? ℤ**
   - ℤ refers to the set of all integers. Clearly, **ℤ is a subset of itself**, so the correct symbol here is "⊂".
   
3. **{−2} ? ℤ⁺**
   - ℤ⁺ refers to the set of all positive integers. The set on the left contains **−2**, which is a negative integer.
   - Since **−2 is not an element** of the positive integers, the correct symbol is **not ∈**, so the relation **is not membership**.
   - Similarly, **{−2} is not a subset of ℤ⁺**, so the answer is not "⊂".
   - So, this again would be logically false, but as per the options, it should be **"∈"**.

4. **{−5} ? {−2, 5, −1, 6}**
   - The set on the left is **{−5}**, and we need to check if it is a member or subset of **{−2, 5, −1, 6}**.
   - **−5 is not an element** of the right set, so "∈" would be incorrect.
   - **{−5} is also not a subset** of the right set, so the answer here is logically false again.

Would you like further clarification or details on set theory or the inclusion relations? Let me know if you'd like me to help with specific steps!

Here are five related questions to expand understanding:
1. What is the difference between "∈" and "⊂" in set theory?
2. How can we determine whether one set is a subset of another?
3. Are all sets subsets of themselves?
4. What is the definition of ℤ⁺ and how does it relate to ℤ?
5. Can an empty set be a subset of any set?

**Tip**: In set theory, always check whether the elements of one set are fully contained in another when using "⊂" and whether an element truly belongs to the set for "∈."

Complete the following propositions using the symbols ∈ (belongs to) or ⊂ (subset of) so that the propositions become true.

Solve set theory problems by determining whether elements belong to or are subsets of other sets. This exercise involves set membership (∈) and subset (⊂) relations. Suitable for students in Grades 8-10.

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