Set Theory Expression Simplification
The question asks us to simplify the set expression: \((A \cup B) - \{(A - B) \cup (B - A) \cup (A \cap B)\}\) Let's break down the expression and simplify it step-by-step.
Understanding Set Operations
We need to understand the basic set operations involved:
- Union (\(A \cup B\)): The set of all elements which are in A, or in B, or in both.
- Difference (\(A - B\)): The set of all elements which are in A but not in B.
- Intersection (\(A \cap B\)): The set of all elements which are common to both A and B.
Simplifying the Inner Expression
Consider the set inside the curly braces: \({(A - B) ∪ (B - A) ∪ (A ∩ B)}\)
- \((A - B)\) represents elements that are only in set A.
- \((B - A)\) represents elements that are only in set B.
- \((A ∩ B)\) represents elements that are in both set A and set B.
When we take the union of these three sets, \((A - B) ∪ (B - A) ∪ (A ∩ B)\), we include elements that are only in A, only in B, and in both A and B. This covers all possible elements belonging to either set A or set B or both.
Therefore, the union of these three parts is exactly the definition of the union of A and B:
\((A - B) \cup (B - A) \cup (A \cap B) = A \cup B\)The term \((A - B) ∪ (B - A)\) is also known as the symmetric difference of A and B, often denoted as \(A \Delta B\). So, the expression can be seen as \((A \Delta B) \cup (A \cap B)\), which also equals \(A \cup B\).
Final Calculation
Now, substitute the simplified inner expression back into the original expression:
\((A \cup B) - \{ A \cup B \}\)We are subtracting the set \((A ∪ B)\) from itself.
Any set subtracted from itself results in the null set (an empty set).
\((A \cup B) - (A \cup B) = \emptyset\)The symbol \(∅\) represents the null set.
Conclusion
The expression \((A ∪ B) - {(A - B) ∪ (B - A) ∪ (A ∩ B)}\) simplifies to the null set.
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