Which of the following is equivalent to A - B̅? A ∪ B, A ∩ B̅, A ∪ B̅, A̅ ∩ B.
Understand the Problem
The question is asking for the set notation that is equivalent to the expression A - B̅. To solve this, we need to understand the relationships between the sets in question and the rules of set operations.
Answer
$A \cap B$
Answer for screen readers
The set notation equivalent to the expression $A - \overline{B}$ is $A \cap B$.
Steps to Solve
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Understand the Symbols
The expression $A - \overline{B}$ indicates the set of elements that are in set $A$ but not in the complement of set $B$. The complement $\overline{B}$ includes all elements not in $B$.
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Define the Complement
The complement of set $B$, denoted as $\overline{B}$, can be described as all elements in the universal set $U$ that are not in $B$:
$$ \overline{B} = U - B $$
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Express the Set Difference
The set difference $A - \overline{B}$ includes all elements in $A$ that are not in $\overline{B}$. Therefore, we can rewrite the expression:
$$ A - \overline{B} = A \cap B $$
This means the elements that are in set $A$ and also in set $B$.
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Conclusion
Therefore, the expression $A - \overline{B}$ simplifies to:
$$ A - \overline{B} = A \cap B $$
The set notation equivalent to the expression $A - \overline{B}$ is $A \cap B$.
More Information
This result shows the intersection of sets $A$ and $B$, demonstrating that you are looking for elements that are common to both sets. Set operations like intersection and complement are foundational concepts within set theory.
Tips
- A common mistake is confusing the complement with the set difference. Remember, set complements include all elements not in the specified set, while the set difference only removes elements of the second set from the first. To avoid confusion, always clarify which set is being complemented.
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