Given two sets A and B, which one of the following is equivalent to A̅ ∪ B̅ ∪ A ∪ B?
Understand the Problem
The question is asking for the equivalent expression of the union of the complements of two sets A and B and the union of the sets A and B themselves. We need to analyze set theory and De Morgan's laws to find the correct equivalent expression.
Answer
$ (A \cap B)' \cup (A \cup B) $
Answer for screen readers
$ (A \cap B)' \cup (A \cup B) $
Steps to Solve
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Identify the Sets and Complements We are given two sets, $A$ and $B$. Their complements are denoted as $A'$ and $B'$.
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State De Morgan's Laws De Morgan's Laws state:
- The complement of a union is the intersection of the complements: $$ (A \cup B)' = A' \cap B' $$
- The complement of an intersection is the union of the complements: $$ (A \cap B)' = A' \cup B' $$
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Express the Given Problem as a Formula We need to express the problem using the union of the complements of $A$ and $B$: The expression we are seeking is: $$ (A' \cup B') \cup (A \cup B) $$
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Apply De Morgan's Laws Using De Morgan’s Laws, we can rewrite the expression $(A' \cup B')$ as: $$ (A \cap B)' $$
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Combine the Expressions Thus, we can write our expression as: $$ (A \cap B)' \cup (A \cup B) $$
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Analyze the Combined Expression This tells us that the final expression is the union of the complement of the intersection and the union of the original sets.
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Simplify if Necessary While we could simplify further depending on the context, this is a standard form to represent the equivalent expression.
$ (A \cap B)' \cup (A \cup B) $
More Information
This answer illustrates a fundamental result in set theory, linking the operations of union and intersection through the complements of sets. Understanding how to apply De Morgan’s Laws is crucial in set theory.
Tips
- Confusing the union and intersection in applying De Morgan's Laws.
- Not correctly identifying the complements of sets.
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