A and B are any two sets, `n(A)=17` and `n(B)=13`. What is the maximum possible value of `n(A uu B)-n(A nn B)` ?

Options

A. 4
B. 13
C. 17
D. 30

Explanation

To solve the problem, we need to find the maximum possible value of \( n(A \cup B) - n(A \cap B) \).

### Step 1: Understand the Sets
We are given two sets, A and B, with the following cardinalities:
- \( n(A) = 17 \)
- \( n(B) = 13 \)

### Step 2: Use the Formula for Union and Intersection
We know the relationship between the union and intersection of two sets:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
From this, we can rearrange the formula to express \( n(A \cup B) - n(A \cap B) \):
\[ n(A \cup B) - n(A \cap B) = n(A) + n(B) - 2n(A \cap B) \]

### Step 3: Substitute the Values
Now, substituting the values of \( n(A) \) and \( n(B) \):
\[ n(A \cup B) - n(A \cap B) = 17 + 13 - 2n(A \cap B) \]
This simplifies to:
\[ n(A \cup B) - n(A \cap B) = 30 - 2n(A \cap B) \]

### Step 4: Maximize the Expression
To maximize \( n(A \cup B) - n(A \cap B) \), we need to minimize \( n(A \cap B) \). The minimum value of \( n(A \cap B) \) is 0, which occurs when sets A and B are disjoint (i.e., they have no elements in common).

### Step 5: Calculate the Maximum Value
If \( n(A \cap B) = 0 \):
\[ n(A \cup B) - n(A \cap B) = 30 - 2(0) = 30 \]

### Final Answer
Thus, the maximum possible value of \( n(A \cup B) - n(A \cap B) \) is **30**.

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