Understanding the Logic Function

The given logic function is $Y = AB + \bar A \bar B$. This expression is equivalent to the XNOR operation between inputs A and B, denoted mathematically as $Y = A \odot B$. The question asks for the minimum number of 2-input NAND gates required to implement this XNOR function.

NAND Gate Implementation Strategy

We need to find the most efficient way to build the XNOR logic using only 2-input NAND gates. It's a known result in digital logic design that the XNOR function requires a specific minimum number of NAND gates.

A standard implementation of the XNOR function ($A \odot B$) using 2-input NAND gates requires exactly 5 gates. The steps are as follows:

1. Implement $N_1 = \text{NAND}(A, B)$.
2. Implement $N_2 = \text{NAND}(N_1, N_1)$. This results in $\text{NOT}(\text{NAND}(A, B)) = \overline{AB} = \bar A + \bar B$.
3. Implement $N_3 = \text{NAND}(A, N_2)$. This results in $\text{NAND}(A, \bar A + \bar B) = \overline{A(\bar A + \bar B)} = \overline{A\bar B}$.
4. Implement $N_4 = \text{NAND}(B, N_2)$. This results in $\text{NAND}(B, \bar A + \bar B) = \overline{B(\bar A + \bar B)} = \overline{\bar A B}$.
5. Implement the final output $Y = \text{NAND}(N_3, N_4)$. This results in $\text{NAND}(\overline{A\bar B}, \overline{\bar A B}) = \overline{(\overline{A\bar B}) \cdot (\overline{\bar A B})} = (\overline{A\bar B}) + (\overline{\bar A B}) = AB + \bar A \bar B$.

This sequence uses exactly 5 distinct 2-input NAND gates.

Minimum Gate Count Justification

The implementation of fundamental logic functions using only 2-input NAND gates follows established minimum counts:

• A NOT gate needs 1 NAND gate.
• An AND or OR gate needs 3 NAND gates.
• An XOR gate needs 4 NAND gates.
• An XNOR gate needs 5 NAND gates.

Since the function $Y = AB + \bar A \bar B$ is the XNOR operation, the theoretical minimum number of 2-input NAND gates required is 5.

Final Answer Determination

Based on the standard implementation and the known minimum counts for logic gates using NAND equivalents, the minimum number of 2-input NAND gates required for the function $Y = AB + \bar A \bar B$ is 5.