Let A and B be sets in a finite universal set U. Given the following: |A – B|, |A ⊕ B|, |A| + |B| and |A ∪ B| Which of the following is in order of increasing size ?
(A) |A – B| < |A ⊕ B| < |A| + |B| < |A ∪ B|
(B) |A ⊕ B| < |A – B| < |A ∪ B| < |A| + |B|
(C) |A ⊕ B| < |A| + |B| < |A – B| < |A ∪ B|
(D) |A – B| < |A ⊕ B| < |A ∪ B| < |A| + |B|
Answer: (D)
Explanation: We will draw venn diagram for all set: |A – B|, |A ⊕ B|, |A| + |B| and |A ∪ B|
So, option (D) is correct.
Alternative way –
|A – B| = |A| - |A ∩ B|
|A ⊕ B| = |A| + |B| - 2|A ∩ B|
|A ∪ B| = |A| + |B| - |A ∩ B|
Therefore,
|A – B| < |A ⊕ B| < |A ∪ B| < |A| + |B|
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