Let \(π΄\) and \(π΅\) be sets in a finite universal set \(π\). Given theβ¦
2016
LetΒ \(π΄\)Β andΒ \(π΅\)Β be sets in a finite universal setΒ \(π\). Given the following :Β \(|A - B|, |A \bigoplus B|, |A| + |B|\) andΒ \(|A \cup B|\) Which of the following is in order of increasing size ?
- A.
\(|A - B| < |A \bigoplus B| < |A| + |B| < |A \cup B|\) - B.
\(|A \bigoplus B| < |A - B| < |A \cup B| < |A| + |B|\) - C.
\(|A \bigoplus B| < |A| + |B| < |A - B| < |A \cup B|\) - D.
\(|A - B| < |A \bigoplus B| < |A \cup B| < |A| + |B|\)
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Correct answer: D
Key facts: use these cardinality identities to compare sizes.
|A β B| = |A \ B| + |B \ A|
|A βͺ B| = |A \ B| + |B \ A| + |A β© B| = |A β B| + |A β© B|
|A| + |B| = |A \ B| + |B \ A| + 2|A β© B| = |A β B| + 2|A β© B|
From these identities we get the chain of inequalities (non-strict):
|A \ B| β€ |A β B|
|A β B| β€ |A βͺ B|
|A βͺ B| β€ |A| + |B|
Combining these gives the overall ordering:
|A \ B| β€ |A β B| β€ |A βͺ B| β€ |A| + |B|
When the sets mentioned on the right-hand sides are nonempty the inequalities become strict. For example, if B \ A is nonempty then |A \ B| < |A β B|; if A β© B is nonempty then |A β B| < |A βͺ B| and |A βͺ B| < |A| + |B|. Therefore the ordering with strict inequalities
|A \ B| < |A β B| < |A βͺ B| < |A| + |B|
is the correct choice in the general case where the added parts are nonempty; otherwise equalities can occur as explained above.
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