Let A = {1, 2, 3, 4}, B = {3, 4}, C = {1, 2, 3, 4, 5},
Z be the set of integers.
Write true or false for each statement and justify your answer.
(a) B ⊂ A (b) C ⊂ A (c) ∅ ⊂ B (d) 1/7∈ Z

Let's analyze each statement and determine whether it is **true** or **false**, providing justifications for each:

### (a) \( B \subset A \)
- **B** is the set \( \{3, 4\} \), and **A** is the set \( \{1, 2, 3, 4\} \).
- A set **B** is a subset of **A** (denoted \( B \subset A \)) if every element of **B** is also in **A**.
- Both elements of **B** (3 and 4) are indeed in **A**.
- Therefore, **B ⊂ A** is **true**.

### (b) \( C \subset A \)
- **C** is the set \( \{1, 2, 3, 4, 5\} \), and **A** is the set \( \{1, 2, 3, 4\} \).
- To be a subset, all elements of **C** must be in **A**.
- However, element 5 is in **C** but not in **A**.
- Therefore, **C ⊄ A**, so this statement is **false**.

### (c) \( \emptyset \subset B \)
- The empty set \( \emptyset \) is considered a subset of **any set**, including **B**.
- This is a fundamental property of sets: the empty set is a subset of every set.
- Therefore, \( \emptyset \subset B \) is **true**.

### (d) \( \frac{1}{7} \in \mathbb{Z} \)
- **Z** represents the set of integers \( \mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\} \).
- \( \frac{1}{7} \) is a rational number, not an integer.
- Therefore, \( \frac{1}{7} \notin \mathbb{Z} \), so this statement is **false**.

### Summary of Answers:
(a) **True**  
(b) **False**  
(c) **True**  
(d) **False**

Would you like further details on any of the justifications?

Here are 5 follow-up questions for practice:
1. What is the difference between a proper subset and a subset?
2. Can the set \( \mathbb{Q} \) (rationals) be a subset of \( \mathbb{Z} \) (integers)?
3. Is \( A \subseteq A \) always true for any set \( A \)?
4. What is the power set of \( A = \{1, 2\} \)?
5. Can an irrational number be a member of \( \mathbb{Z} \)?

**Tip:** Remember, the empty set \( \emptyset \) is always a subset of any set by definition!

Let A = {1, 2, 3, 4}, B = {3, 4}, C = {1, 2, 3, 4, 5}, Z be the set of integers. Write true or false for each statement and justify your answer. (a) B ⊂ A (b) C ⊂ A (c) ∅ ⊂ B (d) 1/7 ∈ Z

Explore a detailed solution to a set theory problem where A = {1, 2, 3, 4}, B = {3, 4}, C = {1, 2, 3, 4, 5}, and Z is the set of integers. Learn about subsets, rational numbers, and the properties of the empty set. Suitable for Grades 8-10.

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