Simplify the Boolean function with don't care conditions and implement it with NAND gates f(w,x,y,z) = (1, 3, 7, 11, 15) d(w,x,y,z) = (0, 2, 5)

Q.) Simplify the Boolean function with don't care conditions and implement it with NAND gates f(w,x,y,z) = (1, 3, 7, 11, 15) d(w,x,y,z) = (0, 2, 5)

 Subject: digital circuit and design

Simplification of Boolean Function:

  1. Karnaugh Map (K-Map) Representation:

    • Construct a 4-variable Karnaugh map for the given function and don't care conditions.
    • Group the 1's and don't care conditions to form the largest possible groups, minimizing the number of literals in each group.

  2. Identification of Prime Implicants:

    • Identify prime implicants, which are minimal sum-of-products expressions, by circling or marking each group of 1's and don't care conditions.

  3. Generation of Essential Prime Implicants:

    • Identify essential prime implicants, which are prime implicants that cover at least one 1 that is not covered by any other prime implicant.

  4. Selection of Prime Implicants for Minimal Expression:

    • Select a minimal set of prime implicants that cover all the 1's in the K-map.
    • This can be done by selecting the essential prime implicants and adding additional prime implicants as needed to cover the remaining 1's.

Simplified Boolean Expression:

The simplified Boolean expression for the given function is:

f(w, x, y, z) = w'z' + x'yz' + y'z

Implementation with NAND Gates:

  1. NAND Gate as Universal Gate:

    • NAND gates are known as universal gates because they can be used to implement any Boolean function.

  2. De Morgan's Theorem Application:

    • Apply De Morgan's theorem to convert the simplified Boolean expression into an equivalent expression using only NAND gates.
    • For example, w'z' can be expressed as NAND(w, z).

  3. NAND Gate Implementation:

    • Construct the circuit diagram by connecting NAND gates according to the simplified Boolean expression.
    • Use additional NAND gates as required to implement the complete circuit.

Final Circuit:

The final circuit using NAND gates for the given Boolean function is shown below:

f(w, x, y, z) = NAND(NAND(w, z), NAND(NAND(x, y), z), NAND(y, z))

This circuit implements the simplified Boolean expression and provides the desired functionality for the given function with don't care conditions.