is the following statement true: if A ∪B ⊂A ∪C, then B ⊂C?

The statement is **not necessarily true**. Here’s why:

The statement \( A \cup B \subset A \cup C \) means that every element in \( A \cup B \) is also in \( A \cup C \). However, this does not imply that \( B \subset C \).

### Explanation
1. \( A \cup B \subset A \cup C \) guarantees that any element in \( B \) that is not in \( A \) must be in \( C \). 
2. However, if an element is in \( B \cap A \) (meaning it is in both \( A \) and \( B \)), then it is automatically in \( A \cup C \), regardless of whether it is in \( C \) or not.
3. Therefore, \( B \) could contain elements that are in \( A \) but not in \( C \), meaning \( B \subset C \) does not necessarily hold.

### Example to Illustrate
Let’s take specific sets:
- Let \( A = \{1\} \),
- Let \( B = \{1, 2\} \),
- Let \( C = \{2\} \).

Here:
- \( A \cup B = \{1, 2\} \),
- \( A \cup C = \{1, 2\} \).

Thus, \( A \cup B = A \cup C \), which satisfies \( A \cup B \subset A \cup C \).
However, \( B = \{1, 2\} \) is **not** a subset of \( C = \{2\} \), since \( 1 \in B \) but \( 1 \notin C \).

### Conclusion
The statement \( A \cup B \subset A \cup C \Rightarrow B \subset C \) is **false**.

Would you like more examples or further clarification?

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1. What conditions make \( B \subset C \) true when \( A \cup B \subset A \cup C \)?
2. Can we find similar statements about intersections, such as \( A \cap B \subset A \cap C \Rightarrow B \subset C \)?
3. How does the power set relate to subset operations like \( A \cup B \subset A \cup C \)?
4. What if \( A \) is an empty set—does it affect whether \( B \subset C \) holds?
5. Are there special cases when \( A \cup B \subset A \cup C \) does imply \( B \subset C \)?

**Tip:** In set theory, verifying logical implications often requires checking if one subset condition can be derived or disproven by counterexamples.

Is the following statement true: if A ∪ B ⊂ A ∪ C, then B ⊂ C?

Explore whether the statement 'if A ∪ B ⊂ A ∪ C, then B ⊂ C' holds true in set theory. This solution examines logical implications and counterexamples to explain why the statement is false, covering key concepts like subsets and union properties. Ideal for high school students in Grades 10-12.

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