How many minimum NAND gate are required to obtain NOR gate :-

Options

A. 3
B. 2
C. 1
D. 4

Answer

To determine how many minimum NAND gates are required to obtain a NOR gate, we can follow these steps:

### Step 1: Understand the Boolean Expressions
- The Boolean expression for a NAND gate is given by:
\[ Y = (A \cdot B)' \]
where \( (A \cdot B)' \) denotes the NOT operation on the AND of A and B.

- The Boolean expression for a NOR gate is given by:
\[ Y = (A + B)' \]
where \( (A + B)' \) denotes the NOT operation on the OR of A and B.

### Step 2: Convert NOR to NAND
To construct a NOR gate using NAND gates, we can use the following logic:
1. **Use two NAND gates as NOT gates**: By connecting both inputs of a NAND gate to the same variable, it behaves as a NOT gate.
- For input A:
\[ Y_1 = (A \cdot A)' = A' \]
- For input B:
\[ Y_2 = (B \cdot B)' = B' \]

2. **Use a third NAND gate to combine the outputs**: The outputs from the first two NAND gates (which act as NOT gates) can be fed into a third NAND gate:
\[ Y_3 = (Y_1 \cdot Y_2)' = (A' \cdot B')' \]
By De Morgan's theorem, this simplifies to:
\[ Y_3 = A + B \]

3. **Final NAND gate for NOR output**: We need to apply NOT again to get the NOR output:
\[ Y = (Y_3)' = (A + B)' \]

### Step 3: Count the NAND Gates
- We used:
- 2 NAND gates to create NOT gates for A and B.
- 1 NAND gate to combine the outputs of the first two.
- 1 final NAND gate to get the NOR output.

Thus, the total number of NAND gates required is:
\[ \text{Total NAND gates} = 2 + 1 + 1 = 4 \]

### Conclusion
Therefore, the minimum number of NAND gates required to obtain a NOR gate is **4**.

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