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:

  1. Given Condition: AB=BA \cup B = B.

    • This means that every element of AA is already in BB, so ABA \subseteq B (i.e., AA is a subset of BB).
  2. Proper Subset Definition:

    • AA is a proper subset of BB (denoted ABA \subset B) if every element of AA is in BB, and there is at least one element in BB that is not in AA.
  3. Implication of the Given Condition:

    • The condition AB=BA \cup B = B does not guarantee that ABA \neq B. If A=BA = B, the union ABA \cup B would still equal BB. Thus, AA might just be equal to BB, not a proper subset.
  4. Conclusion:

    • The statement AB=BA \cup B = B only guarantees ABA \subseteq B, not ABA \subset B (proper subset). Therefore, the statement is false.

Let me know if you want further clarification!
Here are 5 related questions you might find helpful:

  1. What is the difference between a subset and a proper subset?
  2. Can you provide examples where AB=BA \cup B = B but A=BA = B?
  3. How do unions and intersections work in set theory?
  4. What is the definition of set equality?
  5. How does ABA \cap B relate to ABA \cup B?

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