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If X and Y are any two non-empty sets, then what is (X - Y)' equal to?
- a)X' - Y'
- b)X' ∩ Y
- c)X' ∪ Y
- d)X - Y'
Correct answer is option 'C'. Can you explain this answer?
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Understanding Set Difference
When we talk about the expression \(X - Y\), it refers to the set difference between sets \(X\) and \(Y\). This means it includes all elements that are in \(X\) but not in \(Y\).
Explanation of the Options
- Option a: \(X - Y\)
This directly represents the set difference and is not equal to anything else.
- Option b: \(X \cap Y\)
This represents the intersection of sets \(X\) and \(Y\), which consists of elements common to both sets. It does not include elements unique to \(X\).
- Option c: \(X \cup Y\)
This is the union of sets \(X\) and \(Y\), which consists of all elements that are in either set. This includes elements from both sets, not just those in \(X\) but not in \(Y\).
- Option d: \(X - Y\)
This is again the set difference, which is simply a restatement of option a.
Correct Interpretation
The correct answer cannot be \(X - Y\) or \(X \cap Y\) or \(X \cup Y\) in this context. The answer must reflect the unique elements in \(X\) after removing those in \(Y\). Therefore, the correct interpretation among the options provided is that \(X - Y\) simply remains \(X - Y\).
Conclusion
In conclusion, the statement provided seems to mislabel the correct answer. The expression \(X - Y\) is not equal to \(X \cup Y\) or any other option, but rather it is itself. Always remember that \(X - Y\) is unique, representing elements in \(X\) that are not in \(Y\).
When we talk about the expression \(X - Y\), it refers to the set difference between sets \(X\) and \(Y\). This means it includes all elements that are in \(X\) but not in \(Y\).
Explanation of the Options
- Option a: \(X - Y\)
This directly represents the set difference and is not equal to anything else.
- Option b: \(X \cap Y\)
This represents the intersection of sets \(X\) and \(Y\), which consists of elements common to both sets. It does not include elements unique to \(X\).
- Option c: \(X \cup Y\)
This is the union of sets \(X\) and \(Y\), which consists of all elements that are in either set. This includes elements from both sets, not just those in \(X\) but not in \(Y\).
- Option d: \(X - Y\)
This is again the set difference, which is simply a restatement of option a.
Correct Interpretation
The correct answer cannot be \(X - Y\) or \(X \cap Y\) or \(X \cup Y\) in this context. The answer must reflect the unique elements in \(X\) after removing those in \(Y\). Therefore, the correct interpretation among the options provided is that \(X - Y\) simply remains \(X - Y\).
Conclusion
In conclusion, the statement provided seems to mislabel the correct answer. The expression \(X - Y\) is not equal to \(X \cup Y\) or any other option, but rather it is itself. Always remember that \(X - Y\) is unique, representing elements in \(X\) that are not in \(Y\).
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