|X| represent number of elements in region X. Now the following conditions are given
|U|=14, `|(A-B)^(C)|=12, |A cup B|=9 |A triangle B|=7,` Where A and B are two subsets of the universal set U and `A^(c)` represents complement of set A, then

Options

A. |A|=2
B. |B|=5
C. |A|=4
D. |B|=7

Answer

To solve the problem step by step, we will analyze the given conditions and use set theory concepts to find the required values for |A| and |B|.

### Step 1: Understand the Given Information
We have the following information:
- |U| = 14 (the total number of elements in the universal set U)
- |(A - B)^c| = 12 (the number of elements not in A - B)
- |A ∪ B| = 9 (the number of elements in the union of sets A and B)
- |A Δ B| = 7 (the number of elements in the symmetric difference of sets A and B)

### Step 2: Find |A - B|
From the information given, we know that:
\[ |(A - B)^c| = 12 \]
This means that the number of elements in A - B is:
\[ |A - B| = |U| - |(A - B)^c| = 14 - 12 = 2 \]

### Step 3: Use the Union Formula
We know that:
\[ |A ∪ B| = |A| + |B| - |A ∩ B| \]
Let’s denote |A| as a and |B| as b. We can express |A ∪ B| in terms of a and b:
\[ 9 = a + b - |A ∩ B| \tag{1} \]

### Step 4: Use the Symmetric Difference Formula
The symmetric difference |A Δ B| can be expressed as:
\[ |A Δ B| = |A - B| + |B - A| = |A| + |B| - 2|A ∩ B| \]
Substituting the known values:
\[ 7 = 2 + |B - A| \tag{2} \]
From equation (2), we can find |B - A|:
\[ |B - A| = 7 - 2 = 5 \]

### Step 5: Express |B| in Terms of |A ∩ B|
Now, we can express |B| in terms of |A ∩ B|:
Let |A ∩ B| = x. Then:
\[ |B| = |B - A| + |A ∩ B| = 5 + x \]
Substituting this into equation (1):
\[ 9 = a + (5 + x) - x \]
This simplifies to:
\[ 9 = a + 5 \implies a = 4 \]

### Step 6: Find |B|
Now that we have |A| = 4, we can find |B|:
\[ |B| = 5 + x \]
We also know from the union formula:
\[ 9 = 4 + |B| - x \]
Substituting |B|:
\[ 9 = 4 + (5 + x) - x \]
This simplifies to:
\[ 9 = 9 \implies x = |A ∩ B| \text{ can be any value} \]
However, we need to find a specific value. Since |A - B| = 2, we can conclude that:
\[ |A ∩ B| = 2 \]
Thus:
\[ |B| = 5 + 2 = 7 \]

### Final Values
We have:
- |A| = 4
- |B| = 7

### Conclusion
The final answer is:
- |A| = 4
- |B| = 7