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

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

### Explanation:

The statement says that if the union of set \( A \) with set \( B \) is a subset of the union of set \( A \) with set \( C \), then set \( B \) must be a subset of set \( C \).

However, this reasoning is flawed because the set \( A \) could overlap with \( B \) in such a way that elements in \( B \) that are also in \( A \) don't need to be in \( C \) for \( A \cup B \subseteq A \cup C \) to hold. To show that this is false, let's look at a **counterexample**:

### Counterexample:
Let:
- \( A = \{1\} \)
- \( B = \{1, 2\} \)
- \( C = \{3\} \)

Then:
- \( A \cup B = \{1, 2\} \)
- \( A \cup C = \{1, 3\} \)

Clearly, \( A \cup B = \{1, 2\} \subseteq \{1, 3\} = A \cup C \), so the premise \( A \cup B \subseteq A \cup C \) is true.

But \( B = \{1, 2\} \) is **not** a subset of \( C = \{3\} \), since \( 2 \in B \) and \( 2 \notin C \).

Thus, the implication \( B \subseteq C \) does not hold, proving that the original statement is false.

Would you like further clarification on any part of this reasoning?

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Here are 5 related questions to deepen your understanding:
1. What are the properties of set unions and subsets?
2. How can counterexamples be used to disprove logical statements?
3. What is the difference between strict subset and non-strict subset?
4. Can the inclusion \( A \cup B \subseteq A \cup C \) imply something about the relationship between \( B \) and \( C \) in specific cases?
5. How do intersections of sets relate to unions in this context?

**Tip:** When trying to prove or disprove set relationships, constructing counterexamples is a powerful tool for testing the limits of the statement.

Explore the set theory problem: if A ∪ B ⊂ A ∪ C, does it imply that B ⊂ C? This example uses counterexamples to show why this implication is not necessarily true. Ideal for students at a college or advanced high school level.

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