$$ A\cup B=A\cap C\lrArr B\subset A\subset C $$

To analyze the statement $ A \cup B = A \cap C $ and its equivalence to $ B \subset A \subset C $, let's break it down step by step.

1. **Understanding the Original Equation**:
   - $ A \cup B $ is the set of all elements that are in either $ A $ or $ B $ (or both).
   - $ A \cap C $ is the set of all elements that are in both $ A $ and $ C $.

   The equation states that every element in $ A $ or $ B $ is also in both $ A $ and $ C $, and vice versa.

2. **Interpreting the Implication**:
   - The statement $ B \subset A $ means that all elements of $ B $ are also elements of $ A $.
   - The statement $ A \subset C $ means that all elements of $ A $ are also elements of $ C $.

3. **Proof of Equivalence**:
   - **(If)**: Assume $ B \subset A $ and $ A \subset C $.
     - Then, every element of $ B $ is in $ A $ and hence also in $ C $ (since $ A \subset C $).
     - Thus, $ A \cup B = A $ (since $ B $ is contained in $ A $).
     - We have $ A = A \cap C $, which holds true as every element in $ A $ is also in $ C $.
   
   - **(Only If)**: Assume $ A \cup B = A \cap C $.
     - Then, all elements of $ B $ must be in $ A $ (otherwise, $ A \cup B $ would contain elements not in $ A $).
     - Moreover, since $ A \cup B $ is equal to $ A \cap C $, it follows that all elements of $ A $ must be in $ C $ as well.

Thus, the statement $ A \cup B = A \cap C $ is equivalent to $ B \subset A \subset C $.

### Summary
The original equation can be rewritten as the implication that $ B $ is a subset of $ A $, and $ A $ is a subset of $ C $.

If you need any further details or have questions, feel free to ask! Here are some related questions you might consider:

1. What are the definitions of union and intersection in set theory?
2. Can you provide examples of sets $ A $, $ B $, and $ C $ that satisfy this equivalence?
3. How would this change if we removed the subset conditions?
4. What happens if $ A $ and $ C $ are equal sets?
5. How can we visualize this relationship with Venn diagrams?

**Tip**: When working with set relations, drawing a Venn diagram can often clarify how sets overlap and relate to each other.

Explore the relationship between the sets A, B, and C, examining the equivalence of A ∪ B = A ∩ C with B ⊆ A ⊆ C. This problem covers essential set theory concepts, including union, intersection, and subset relations, suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation