https://file.aiquickdraw.com/mathbot/file/1726707063612icn1mu6r.jpg

Let's break down the questions in the image and provide solutions for both sections:

### A. True or False (Write T for True and F for False)

1. **_____ Subsets are the objects or ideas in a set.**
   - **Answer**: True (T). Subsets contain elements that are part of the original set.

2. **_____ If a set is empty, then the number of its subset is 0.**
   - **Answer**: False (F). An empty set has exactly one subset, which is the empty set itself.

3. **_____ An empty set is a finite set.**
   - **Answer**: True (T). The empty set is considered finite because it contains no elements.

4. **_____ Let A = {1, 2, 3} and B = {2, 3, 4}. Then {1, 2} is a subset of A-B.**
   - **Answer**: False (F). A - B = {1}, and {1, 2} is not a subset of {1}.

5. **_____ {a, b} ⊂ {b, a}**
   - **Answer**: False (F). Sets are unordered, so {a, b} = {b, a}, meaning they are equal, not a subset.

6. **_____ {a} ∈ {b, a, d, f}**
   - **Answer**: False (F). The set {a} is not an element; 'a' is.

7. **_____ {0} ⊆ ∅**
   - **Answer**: False (F). {0} is not a subset of the empty set because the empty set contains no elements.

8. **_____ Let {a, b, c}. Then a, b ⊆ A.**
   - **Answer**: False (F). Elements a and b are part of the set, but they are not subsets.

9. **_____ {x | x ∈ ℕ and x > 10} ⊂ ℕ**
   - **Answer**: True (T). The set of natural numbers greater than 10 is a subset of all natural numbers.

10. **_____ If a set is a singleton set, then the number of its subset will be 1.**
   - **Answer**: False (F). A singleton set {x} has two subsets: the empty set and {x} itself.

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### B. Set Operations (Let A, B, C ⊂ U, where U is the set of the twelve months in the Gregorian calendar)
- Set A: months starting with J = {January, June, July}
- Set B: months with exactly five letters = {March, April, May, June, July}
- Set C: months starting with M = {March, May}

#### Find the following:

1. **The Universal set U**
   - **Answer**: U = {January, February, March, April, May, June, July, August, September, October, November, December}

2. **A ∩ B**
   - **Answer**: A ∩ B = {June, July} (months in both A and B)

3. **A ∩ C**
   - **Answer**: A ∩ C = ∅ (no month starts with J and M simultaneously)

4. **B ∩ C**
   - **Answer**: B ∩ C = {March, May} (common months between B and C)

5. **(A - C) ∩ (B - C)**
   - **Answer**: (A - C) = {January, June, July}, (B - C) = {April, June, July}, so (A - C) ∩ (B - C) = {June, July}

6. **(B - C)'**
   - **Answer**: Complement of (B - C) = U - {April, June, July} = {January, February, March, May, August, September, October, November, December}

7. **((B - C)')'**
   - **Answer**: ((B - C)')' = B - C = {April, June, July}

8. **A - B'**
   - **Answer**: B' = U - B = {January, February, August, September, October, November, December}, A - B' = {June, July}

9. **(B - C)'**
   - **Answer**: This is already answered in #6: (B - C)' = {January, February, March, May, August, September, October, November, December}

10. **(B - C) - B**
    - **Answer**: B - C = {April, June, July}, (B - C) - B = ∅ (no elements left after subtracting B)

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Do you have any questions or would you like further details? 

Here are some follow-up questions to expand your understanding:
1. What are the properties of subsets and how are they different from elements?
2. How is the concept of a singleton set important in set theory?
3. Can you list all possible subsets of a set containing three elements?
4. How does the complement operation work in set theory?
5. What is the significance of the universal set in real-world applications?

**Tip**: When working with sets, always pay close attention to whether the problem is asking for elements (∈) or subsets (⊆), as this is a common source of confusion.

Activity 2 involving True/False questions and set operations as shown in the image.

Explore the fundamental concepts of set theory through Activity 2, which includes True/False questions and set operations. This activity covers key topics such as properties of subsets, set operations, and the use of Venn diagrams. Suitable for high school students studying mathematics.

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