Question

If \(|A| = 25, |B| = 30, |C| = 45, |A \cap B| = 5, |B \cap C| = 10, |A \cap C| = 6, |A \cap B \cap C| = 2\) then arrange the following in non-decreasing order:
A. \(|A \cup B|\)
B. \(|B \cup C|\)
C. \(|A \cup B \cup C|\)
D. \(|C \cup A|\)
E. \(|A - B|\)
Choose the correct answer from the options given below:

• A. C, B, A, E, D
• B. E, A, D, C, B
• C. E, A, D, B, C
• D. E, D, A, B, C

Hint:

We know that \(|A-B| = 20\) must be the smallest value as it is only a subset of \(A\) (size 25).
Also, the union of all three sets, \(|A \cup B \cup C| = 81\), must logically be the largest value.
Thus, the sequence must begin with E and end with C.
Looking at the options, only (C) and (D) satisfy this, and verifying the order of D (64) < B (65) points uniquely to (C).

Answer 1

The Correct Option is C

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Step 1: Understanding the Question: This question is based on set operations and cardinality.
We are given the individual sizes of three sets and their intersections.
We need to compute the cardinalities for several set combinations and arrange them in non-decreasing (ascending) order.

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Step 2: Key Formula or Approach:
1. Cardinality of union of two sets:
\[ |X \cup Y| = |X| + |Y| - |X \cap Y| \]
2. Cardinality of union of three sets:
\[ |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |A \cap C| + |A \cap B \cap C| \]
3. Cardinality of set difference:
\[ |A - B| = |A| - |A \cap B| \]

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Step 3: Detailed Explanation:
Let us calculate the value for each item:
- Calculation of A: \(|A \cup B|\)
\[ |A \cup B| = |A| + |B| - |A \cap B| \]
\[ |A \cup B| = 25 + 30 - 5 = 50 \]
- Calculation of B: \(|B \cup C|\)
\[ |B \cup C| = |B| + |C| - |B \cap C| \]
\[ |B \cup C| = 30 + 45 - 10 = 65 \]
- Calculation of C: \(|A \cup B \cup C|\)
\[ |A \cup B \cup C| = 25 + 30 + 45 - 5 - 10 - 6 + 2 \]
\[ |A \cup B \cup C| = 100 - 21 + 2 = 81 \]
- Calculation of D: \(|C \cup A|\)
\[ |C \cup A| = |C| + |A| - |C \cap A| \]
\[ |C \cup A| = 45 + 25 - 6 = 64 \]
- Calculation of E: \(|A - B|\)
\[ |A - B| = |A| - |A \cap B| \]
\[ |A - B| = 25 - 5 = 20 \]
Summarizing the calculated values:
- A = 50
- B = 65
- C = 81
- D = 64
- E = 20
Now, let us arrange these values in ascending (non-decreasing) order:
\[ 20 < 50 < 64 < 65 < 81 \]
Which corresponds to the order:
\[ E \rightarrow A \rightarrow D \rightarrow B \rightarrow C \]

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Step 4: Final Answer: The correct non-decreasing order is E, A, D, B, C, matching option (C).