How many minimum NAND GATES are required for obtaining an output of A. B + C. D ?

Explanation

To determine the minimum number of NAND gates required to obtain the output of \( A \cdot B + C \cdot D \), we can follow these steps:

### Step 1: Understand the Expression
The expression \( A \cdot B + C \cdot D \) is in the Sum of Products (SOP) form. This means we need to find a way to implement this expression using NAND gates.

### Step 2: Apply De Morgan's Theorem
Using De Morgan's theorem, we can rewrite the expression in a form suitable for NAND gates. According to De Morgan's theorem:
\[ X + Y = (X' \cdot Y')' \]
Thus, we can express \( A \cdot B + C \cdot D \) as:
\[ (A \cdot B + C \cdot D)'' \]
This means we will first find \( A \cdot B \) and \( C \cdot D \), and then apply the NAND operation on the results.

### Step 3: Implementing the Terms
1. **First NAND gate**: To find \( A \cdot B \), we can use a NAND gate. The output of this gate will be \( (A \cdot B)' \).
- Inputs: A, B
- Output: \( (A \cdot B)' \)

2. **Second NAND gate**: To find \( C \cdot D \), we can use another NAND gate. The output will be \( (C \cdot D)' \).
- Inputs: C, D
- Output: \( (C \cdot D)' \)

### Step 4: Combine the Results
3. **Third NAND gate**: Now, we need to combine the outputs of the first two NAND gates. We will take the outputs \( (A \cdot B)' \) and \( (C \cdot D)' \) and apply a NAND operation to them.
- Inputs: \( (A \cdot B)' \), \( (C \cdot D)' \)
- Output: \( ((A \cdot B)' \cdot (C \cdot D)')' \) which is equivalent to \( A \cdot B + C \cdot D \).

### Conclusion
Thus, the minimum number of NAND gates required to implement the expression \( A \cdot B + C \cdot D \) is **3**.