Math Problem Statement
Determine if the statement below is true or false.
If
Upper A union Upper BA∪Bequals=Upper BB,
then
Upper A is a proper subset of Upper BA⊂B.
Question content area bottom
Part 1
Is the statement true or false?
True
TrueTrue
False
False
Solution
The statement is false. Here's the reasoning:
Explanation:
-
Given Condition: .
- This means that every element of is already in , so (i.e., is a subset of ).
-
Proper Subset Definition:
- is a proper subset of (denoted ) if every element of is in , and there is at least one element in that is not in .
-
Implication of the Given Condition:
- The condition does not guarantee that . If , the union would still equal . Thus, might just be equal to , not a proper subset.
-
Conclusion:
- The statement only guarantees , not (proper subset). Therefore, the statement is false.
Let me know if you want further clarification!
Here are 5 related questions you might find helpful:
- What is the difference between a subset and a proper subset?
- Can you provide examples where but ?
- How do unions and intersections work in set theory?
- What is the definition of set equality?
- How does relate to ?
Tip: When dealing with set relationships, always consider both inclusion and equality to distinguish between subsets and proper subsets.
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Math Problem Analysis
Mathematical Concepts
Set Theory
Subset and Proper Subset
Formulas
Union of Sets (A ∪ B)
Definition of Proper Subset (A ⊂ B)
Theorems
Subset Definition (A ⊆ B)
Union Property
Suitable Grade Level
Grades 9-12