Write the minterm of ACD + AB + AC and implement it using NAND gates.

To write the minterm of the given expression ACD + AB + AC and implement it using NAND gates, we will follow these steps:

Step 1: Write the Expression in Canonical Form

The given expression is ACD + AB + AC. To write it in canonical form, we need to include all the missing variables in each term so that each term becomes a minterm. A minterm is a product (AND) of all the variables in the system, either in true form or complemented form.

Assuming we have a three-variable system (A, B, C), the canonical form of the given expression would be:

• ACD: This term already includes all variables, so it's a minterm.
• AB: This term is missing the variable C. To make it a minterm, we need to include both C and its complement (C + C'). So the term becomes AB(C + C').
• AC: This term is missing the variable B. To make it a minterm, we need to include both B and its complement (B + B'). So the term becomes AC(B + B').

Now, let's expand the terms AB(C + C') and AC(B + B'):

• AB(C + C') = ABC + ABC'
• AC(B + B') = ABC + ABC'

Now, the expression in canonical form (sum of minterms) is:

ACD + ABC + ABC' + ABC + ABC'

Since ABC appears twice, we can simplify the expression by removing the duplicate:

ACD + ABC + ABC'

Step 2: Create the Truth Table

To visualize the minterms, let's create a truth table for the variables A, B, and C:

A	B	C	ACD	AB	AC	Expression
0	0	0	0	0	0	0
0	0	1	0	0	0	0
0	1	0	0	0	0	0
0	1	1	0	0	0	0
1	0	0	0	0	0	0
1	0	1	1	0	1	1
1	1	0	0	1	1	1
1	1	1	1	1	1	1

The last column represents the output of the expression ACD + ABC + ABC'. We can see that the output is 1 for the minterms ABCD (110), ABCD' (101), and ABC'D (111).

Step 3: Implement Using NAND Gates

To implement the expression using NAND gates, we need to apply De Morgan's theorem to convert the expression into a form that only uses NAND operations. The NAND gate is a universal gate, which means we can create any logical function using only NAND gates.

The given expression in sum of products form is:

[ \text{Expression} = ACD + ABC + ABC' ]

To convert this into NAND-only form, we first need to realize that a NAND gate is equivalent to an AND gate followed by a NOT gate. Therefore, we can convert the OR operations into NAND operations by applying De Morgan's theorem:

[ \overline{\overline{ACD} \cdot \overline{ABC} \cdot \overline{ABC'}} ]

Now, let's implement this using NAND gates:

1. Create the NAND equivalent of ACD by NANDing A, C, and D together.
2. Create the NAND equivalent of ABC by NANDing A, B, and C together.
3. Create the NAND equivalent of ABC' by NANDing A, B, and the complement of C together.
4. Finally, NAND the results of the above three operations together.

Here's a diagram representing the implementation using NAND gates:

 A---|    |---|    |
     |NAND|   |    |---|    |
 C---|    |---|NAND|   |    |---|    |
     |    |   |    |---|NAND|   |    |
 D---|    |   |    |   |    |---|NAND|--- Output
     |    |   |    |   |    |
 A---|    |   |    |   |    |
     |NAND|   |    |   |    |
 B---|    |---|NAND|   |    |
     |    |   |    |---|    |
 C---|    |   |    |   |    |
     |    |   |    |   |    |
 A---|    |   |    |   |    |
     |NAND|   |    |   |    |
 B---|    |---|NAND|   |    |
     |    |   |    |---|    |
C'--|NOT |   |    |   |    |

Note that the NOT gate for C' is also implemented using a NAND gate by connecting both inputs of the NAND gate to C.

This diagram represents the logical implementation of the given expression using only NAND gates. Each NAND gate performs the operation of combining the inputs and inverting the result. The final NAND gate inverts the result of the AND operations, which is equivalent to the OR operation in the original expression due to De Morgan's theorem.