Let A and B be two subsets of the universal set U. Which of the following is equivalent to A ∪ (A ∩ B)̅ ?
Understand the Problem
The question is asking for an equivalent expression for the union of set A and the complement of the intersection of sets A and B within the universal set U. We need to analyze the set operations and simplify the expression to find which of the provided options is equivalent.
Answer
$U$
Answer for screen readers
The equivalent expression is $U$.
Steps to Solve
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Identify the Sets and Operations First, we identify what we need. We have set $A$, set $B$, and the universal set $U$. We are looking for the expression for $A \cup (A \cap B)'$, where $(A \cap B)'$ is the complement of the intersection of sets $A$ and $B$.
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Expressing the Complement The complement of the intersection $A \cap B$ can be expressed within the universal set $U$: $$(A \cap B)' = U - (A \cap B)$$
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Using De Morgan's Law According to De Morgan's Law, the complement of the intersection can also be rewritten as: $$(A \cap B)' = A' \cup B'$$
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Combining the Sets Now we substitute this into our original expression: $$A \cup (A \cap B)' = A \cup (A' \cup B')$$
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Applying Associative Property We can rearrange the sets using the associative property of unions: $$= (A \cup A') \cup B'$$
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Simplifying the Union The union of a set and its complement is the universal set: $$A \cup A' = U$$
Now we continue: $$= U \cup B'$$
- Final Result The union of the universal set with any set is still the universal set: $$U \cup B' = U$$
The equivalent expression is $U$.
More Information
The result shows that when combining a set with the complement of its intersection with another set within the universal set, we ultimately cover the entire universal set itself. This is a valuable property in set theory and is rooted in the principles of set operations.
Tips
- Forgetting that the union of a set and its complement equals the universal set.
- Misapplying De Morgan's Law by incorrectly interpreting intersections and unions.
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