Math Problem Statement

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

Solution

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

The statement ABACA \cup B \subset A \cup C means that every element in ABA \cup B is also in ACA \cup C. However, this does not imply that BCB \subset C.

Explanation

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

Example to Illustrate

Let’s take specific sets:

  • Let A={1}A = \{1\},
  • Let B={1,2}B = \{1, 2\},
  • Let C={2}C = \{2\}.

Here:

  • AB={1,2}A \cup B = \{1, 2\},
  • AC={1,2}A \cup C = \{1, 2\}.

Thus, AB=ACA \cup B = A \cup C, which satisfies ABACA \cup B \subset A \cup C. However, B={1,2}B = \{1, 2\} is not a subset of C={2}C = \{2\}, since 1B1 \in B but 1C1 \notin C.

Conclusion

The statement ABACBCA \cup B \subset A \cup C \Rightarrow B \subset C is false.

Would you like more examples or further clarification?


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

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

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Math Problem Analysis

Mathematical Concepts

Set Theory
Subsets
Union of Sets
Logical Implications

Formulas

A ∪ B ⊂ A ∪ C

Theorems

Subset and Union Properties
Counterexample for Subset Implication

Suitable Grade Level

Grades 10-12