Implement Y = AB + CD using only NAND gates.

To implement the logical function Y = AB + CD using only NAND gates, we need to understand the properties of NAND gates and how they can be used to create other logical functions. A NAND gate is a universal gate, which means that you can create any other logical operation using just NAND gates. The truth table for a NAND gate is as follows:

A	B	A NAND B
0	0	1
0	1	1
1	0	1
1	1	0

The logical expression for a NAND gate is:

[ \overline{AB} ]

Now, let's break down the process of implementing Y = AB + CD using only NAND gates step by step:

Step 1: Express the function using NAND operations

We can use De Morgan's Theorem to express the OR operation in terms of NAND operations. De Morgan's Theorem states that:

[ \overline{A + B} = \overline{A} \cdot \overline{B} ]

Using this theorem, we can rewrite the original function Y = AB + CD as:

[ Y = \overline{\overline{AB + CD}} ]

Now, we apply De Morgan's Theorem to the inner expression:

[ Y = \overline{\overline{AB} \cdot \overline{CD}} ]

Step 2: Implement the function using NAND gates

Now that we have the function expressed in terms of NAND operations, we can implement it using NAND gates. Here's how we can do it:

1. Create the AB term using a NAND gate, but since we need AB and not its negation, we need to negate the output of the NAND gate. This requires two NAND gates: one to create the (\overline{AB}) term and another to negate it to get AB.

2. Similarly, create the CD term using two NAND gates: one to create the (\overline{CD}) term and another to negate it to get CD.

3. Finally, use a NAND gate to combine the negations of the AB and CD terms to get the final output Y.

Step 3: Draw the circuit diagram

Here's the circuit diagram using only NAND gates to implement Y = AB + CD:

  A ----|    |----|    |
        |NAND|    |NAND|----|    |
  B ----|    |----|    |    |NAND|---- Y
                    |  |    |
  C ----|    |----|  |  |    |
        |NAND|    |  |  |    |
  D ----|    |----|  |  |    |
                    |  |    |
                    |  |    |
                    |  |    |
                    |  |    |
                    ----    |
                             |
                             |

Step 4: Verify the implementation

To verify that the implementation is correct, we can create a truth table for the circuit and compare it to the truth table for the original function Y = AB + CD.

A	B	C	D	AB	CD	(\overline{AB})	(\overline{CD})	Y = AB + CD	NAND Implementation
0	0	0	0	0	0	1	1	0	0
0	0	0	1	0	0	1	1	0	0
0	0	1	0	0	0	1	1	0	0
0	0	1	1	0	1	1	0	1	1
0	1	0	0	0	0	1	1	0	0
0	1	0	1	0	0	1	1	0	0
0	1	1	0	0	0	1	1	0	0
0	1	1	1	0	1	1	0	1	1
1	0	0	0	0	0	1	1	0	0
1	0	0	1	0	0	1	1	0	0
1	0	1	0	0	0	1	1	0	0
1	0	1	1	0	1	1	0	1	1
1	1	0	0	1	0	0	1	1	1
1	1	0	1	1	0	0	1	1	1
1	1	1	0	1	0	0	1	1	1
1	1	1	1	1	1	0	0	1	1

As we can see from the truth table, the output of the NAND implementation matches the output of the original function Y = AB + CD for all possible input combinations. This confirms that the implementation using only NAND gates is correct.