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 means that every element in is also in . However, this does not imply that .
Explanation
- guarantees that any element in that is not in must be in .
- However, if an element is in (meaning it is in both and ), then it is automatically in , regardless of whether it is in or not.
- Therefore, could contain elements that are in but not in , meaning does not necessarily hold.
Example to Illustrate
Let’s take specific sets:
- Let ,
- Let ,
- Let .
Here:
- ,
- .
Thus, , which satisfies . However, is not a subset of , since but .
Conclusion
The statement is false.
Would you like more examples or further clarification?
- What conditions make true when ?
- Can we find similar statements about intersections, such as ?
- How does the power set relate to subset operations like ?
- What if is an empty set—does it affect whether holds?
- Are there special cases when does imply ?
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