Let A and B be sets, prove B \subset A \cup B

To prove that \( B \subset A \cup B \), we need to show that every element of \( B \) is also an element of \( A \cup B \).

### Proof:

1. **Let \( x \in B \)**:  
   This means that \( x \) is an element of the set \( B \).

2. **Definition of Union**:  
   The set \( A \cup B \) consists of all elements that are either in \( A \) or in \( B \) (or both). Formally:  
   \[
   A \cup B = \{ x \mid x \in A \text{ or } x \in B \}
   \]

3. **Since \( x \in B \)**, by the definition of union, we have:
   \[
   x \in A \cup B
   \]
   because if \( x \in B \), it satisfies the condition to be in \( A \cup B \).

4. **Conclusion**:  
   Since \( x \in B \) implies \( x \in A \cup B \), it follows that every element of \( B \) is also an element of \( A \cup B \). Therefore, \( B \subset A \cup B \).

### Final statement:
Thus, we have shown that \( B \subset A \cup B \).

---

Would you like further details or have any questions?

Here are five related questions to deepen your understanding:
1. What is the difference between \( A \subset B \) and \( A = B \)?
2. Can you find a counterexample where \( B \subset A \) does not hold?
3. What does \( A \cap B \) represent and how is it different from \( A \cup B \)?
4. How do we prove that \( A \cup B \subset A \cup B \) is always true?
5. What are the set identities involving union and intersection?

**Tip**: Always remember that proving set inclusion involves showing that every element of one set belongs to the other!

Explore the step-by-step proof showing that for any sets A and B, the set B is a subset of A ∪ B. This solution covers key concepts in set theory such as set inclusion and union. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation