Math Problem Statement
True or false
(A ∪ B')' = A' ∩ B
Solution
The given equation is:
This equation represents a logical statement involving set operations, specifically unions, complements, and intersections. Let's break it down to determine if it's true or false.
Left-hand side:
- means the union of set and the complement of set .
- refers to the complement of the union of and . By De Morgan's Law, the complement of a union is the intersection of the complements: (Since ).
Right-hand side:
This is exactly the same expression as the simplified left-hand side: .
Conclusion:
Both sides of the equation are identical. Therefore, the statement is true.
Would you like more details or have any other questions? Here are 5 related questions to explore this topic further:
- What is De Morgan's law in set theory?
- How do you find the complement of a set?
- What is the difference between union and intersection of sets?
- Can De Morgan's law be applied to more than two sets?
- How do Venn diagrams visually represent set operations?
Tip: When dealing with set operations, using Venn diagrams can help visualize complex relationships and verify results.
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Math Problem Analysis
Mathematical Concepts
Set Theory
De Morgan's Laws
Complement
Union
Intersection
Formulas
(A ∪ B')' = A' ∩ B
Theorems
De Morgan's Law
Suitable Grade Level
Grades 9-12