If \(A\) and \(B\) are non-empty subsets of a set, and \(A^c\) and \(B^c\) represent their complements, then which of the following is/are correct? I. \(A-B = B^c - A^c\) II. \(A - B^c = A^c - B\) Select the answer using the code given below.
Analyzing Set Theory Identities: Complements and Set Difference
This solution examines two statements involving set differences and complements to determine their correctness.
Statement I: Verifying \(A - B = Bc - Ac\)
Let's analyze the first statement: \(A - B = Bc - Ac\).
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Definition of Set Difference: The set difference \(A - B\) contains all elements that are in set \(A\) but not in set \(B\). Mathematically, \(A - B = {x | x ∈ A and x ∉ B}\).
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Analyzing the Left Side: The left side is \(A - B\), which represents elements in \(A\) and not in \(B\).
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Analyzing the Right Side: The right side is \(Bc - Ac\).
- Using the definition of set difference, \(Bc - Ac = {x | x ∈ Bc and x ∉ Ac}\).
- Recall the definition of a complement: \(x ∈ Bc\) means \(x\) is not in \(B\) (\(x ∉ B\)).
- Also, \(x ∉ Ac\) means \(x\) is in \(A\) (\(x ∈ A\)).
- Substituting these back, we get: \(Bc - Ac = {x | x ∉ B and x ∈ A}\).
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Comparison: We see that \({x | x ∈ A and x ∉ B}\) is exactly the same as \({x | x ∉ B and x ∈ A}\). Therefore, the left side \(A - B\) is equal to the right side \(Bc - Ac\).
Conclusion for Statement I: Statement I is correct.
Statement II: Verifying \(A - Bc = Ac - B\)
Now, let's analyze the second statement: \(A - Bc = Ac - B\).
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Analyzing the Left Side: \(A - Bc\).
- Using the definition of set difference: \(A - Bc = {x | x ∈ A and x ∉ Bc}\).
- Since \(x ∉ Bc\) means \(x ∈ B\), we can rewrite this as: \(A - Bc = {x | x ∈ A and x ∈ B}\).
- This is the definition of the intersection of \(A\) and \(B\), denoted as \(A ∩ B\).
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Analyzing the Right Side: \(Ac - B\).
- Using the definition of set difference: \(Ac - B = {x | x ∈ Ac and x ∉ B}\).
- Since \(x ∈ Ac\) means \(x ∉ A\), we can rewrite this as: \(Ac - B = {x | x ∉ A and x ∉ B}\).
- This represents elements that are in neither \(A\) nor \(B\).
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Comparison: We found that the left side equals \(A ∩ B\) and the right side equals \({x | x ∉ A and x ∉ B}\). These two sets are generally not equal. For example, elements in the intersection (\(A ∩ B\)) are in both \(A\) and \(B\), while the elements described by the right side are in neither.
Counterexample: Let the universal set be \(U = {1, 2, 3, 4}\), \(A = {1, 2}\), and \(B = {2, 3}\). Then \(Ac = {3, 4}\) and \(Bc = {1, 4}\).
- Left side: \(A - Bc = {1, 2} - {1, 4} = {2}\).
- Right side: \(Ac - B = {3, 4} - {2, 3} = {4}\).
- Since \({2} ≠ {4}\), the statement is false.
Conclusion for Statement II: Statement II is incorrect.
Final Conclusion
Based on the analysis of both statements:
- Statement I (\(A - B = Bc - Ac\)) is correct.
- Statement II (\(A - Bc = Ac - B\)) is incorrect.
Therefore, only Statement I is correct.
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