Which of the following is equivalent to A - B? A ⊕ B, A ∩ B, A ∪ B, A ∩ B̅
Understand the Problem
The question is asking which of the mathematical expressions given is equivalent to the expression A - B. This requires an understanding of set operations such as intersection, union, and the complement of sets.
Answer
$A \cap B'$
Answer for screen readers
The expression equivalent to $A - B$ is $A \cap B'$.
Steps to Solve
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Identify the Expression We start with the expression we need to find an equivalent for, which is $A - B$. This symbolizes the elements in set $A$ that are not in set $B$.
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Use Set Operations Understand that the expression $A - B$ can be represented using other set operations. The difference $A - B$ is equivalent to the intersection of $A$ with the complement of $B$: $$ A - B = A \cap B' $$
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Explain the Complement The complement of set $B$, denoted $B'$, includes all elements that are not in $B$. Thus, the expression could also be visualized as selecting all elements in $A$ while excluding those in $B$.
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Summarize the Relationships To ensure clarity, remember that:
- $A - B$ includes all elements in $A$ excluding those found in $B$.
- This is equivalent to the intersection of $A$ with elements outside of $B$, formally expressed as $A \cap B'$.
The expression equivalent to $A - B$ is $A \cap B'$.
More Information
This expression utilizes basic set theory concepts such as set operations. Understanding these concepts can help clarify many problems involving sets and their relationships.
Tips
- A common mistake is confusing $A - B$ with $A \cap B$, which includes elements that are in both $A$ and $B$ instead.
- Another mistake is mixing up the notation: ensure that the complement is correctly represented as $B'$ and not just $B$.
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