Simplify the Boolean function using K-map F(A, B, C, D) = ∑ (0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 14)

Q.) Simplify the Boolean function using K-map F(A, B, C, D) = ∑ (0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 14)

 Subject: Digital Circuit Design

The Karnaugh map (K-map) is a method of simplifying Boolean algebra expressions. The K-map can take two forms, a two-dimensional (2-D) table or a four-dimensional (4-D) cube. The K-map method of simplifying Boolean expressions is graphical in nature, and it is an alternative to the algebraic simplification method.

The given Boolean function is:

F(A, B, C, D) = ∑ (0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 14)

This function can be simplified using a 4-variable K-map as follows:

Step 1: Draw a 4-variable K-map. The K-map for a 4-variable Boolean function is a 4x4 grid. The variables A and B represent the rows, and the variables C and D represent the columns.

CD\AB 00 01 11 10
00
01
11
10

Step 2: Fill in the K-map. The given function tells us that the output is 1 for the minterms 0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 14. We fill in these minterms in the K-map with 1s. The rest of the cells are filled with 0s.

CD\AB 00 01 11 10
00 1 1 0 1
01 1 1 0 1
11 0 1 1 1
10 1 1 1 0

Step 3: Group the 1s. The next step is to group the 1s in the K-map. The groups should be as large as possible, and each group should contain a number of cells that is a power of 2 (1, 2, 4, 8, etc.). Each 1 must be part of at least one group.

In this case, we can make three groups:

  • Group 1: The four 1s in the middle of the K-map.
  • Group 2: The two 1s in the top row.
  • Group 3: The two 1s in the bottom row.

Step 4: Write the simplified Boolean function. Each group of 1s corresponds to a term in the simplified Boolean function. The variables in each term are determined by the row and column labels of the cells in the group. If a variable changes within a group, it is not included in the corresponding term.

  • Group 1 corresponds to the term BC.
  • Group 2 corresponds to the term A' D'.
  • Group 3 corresponds to the term AD.

So, the simplified Boolean function is:

F(A, B, C, D) = BC + A' D' + AD

This is the simplified form of the given Boolean function using the K-map method.