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

No, the statement "if \( A \cup B \subset A \cup C \), then \( B \subset C \)" is **not always true**. 

### Counterexample:
Consider the following sets:
- \( A = \{1\} \)
- \( B = \{2\} \)
- \( C = \{3\} \)

Now, we check the union and subset relations:
- \( A \cup B = \{1, 2\} \)
- \( A \cup C = \{1, 3\} \)

Clearly, \( A \cup B = \{1, 2\} \subset A \cup C = \{1, 3\} \) is **false**, and \( B = \{2\} \not\subset C = \{3\} \), so the initial statement doesn't hold for these specific sets.

### Explanation:
The condition \( A \cup B \subset A \cup C \) only means that every element in \( B \) is either in \( C \) or in \( A \). It does not necessarily imply that \( B \subset C \). The element of \( B \) might be present in \( A \), making the inclusion \( A \cup B \subset A \cup C \) true without requiring \( B \) to be a subset of \( C \).

### Would you like further details or another example? 

Here are some related questions:

1. What are the conditions for \( A \cup B \subset A \cup C \) to hold?
2. Can you explain the relationship between \( B \) and \( C \) when \( A \cup B \subset A \cup C \)?
3. How can you prove a counterexample using different sets for \( A \), \( B \), and \( C \)?
4. What is the difference between the union and subset operations in set theory?
5. Can you apply the same logic to intersections of sets?

**Tip:** Always check set inclusions carefully by testing small examples to see if a general statement holds.

Explore whether the set relationship A ∪ B ⊂ A ∪ C implies B ⊂ C with a counterexample and detailed explanation. Understand key concepts in set theory, including union and subset operations. Suitable for students in Grades 10-12.

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