Boolean Function Simplification Using Karnaugh Map

To simplify the given Boolean function \(F(a, b, c) = \sum(0, 2, 4, 5, 6)\), we can use a Karnaugh Map (K-map). The K-map is a visual method used to simplify Boolean expressions by grouping adjacent minterms (product terms where all variables appear once, either in true or complemented form).

Boolean Minterm Representation

First, let's list the minterms corresponding to the given sum:

• m0: \(000 \implies a'b'c'\)
• m2: \(010 \implies a'bc'\)
• m4: \(100 \implies ab'c'\)
• m5: \(101 \implies ab'c\)
• m6: \(110 \implies abc'\)

We will place a '1' in the corresponding cells of the 3-variable K-map for these minterms, and '0's for the remaining minterms.

Karnaugh Map Construction for F(a,b,c)

For a 3-variable function \(F(a, b, c)\), the K-map will have \(2^3 = 8\) cells. We arrange these cells in a Gray code sequence to ensure that adjacent cells differ by only one bit. The rows represent variable 'a' and columns represent 'bc'.

Grouping Minterms for Simplification

After placing the '1's, the next step is to group adjacent '1's in powers of two (2, 4, 8, etc.). The goal is to form the largest possible groups to achieve maximum simplification.

1. Group 1 (Four 1's):
   • Observe the '1's at positions (0, 0), (0, 2), (1, 0), and (1, 2) in the map (corresponding to minterms m0, m2, m4, m6).
   • These four cells (000, 010, 100, 110) are adjacent and form a group of four.
   • In this group, variable 'c' is always '0' (\(c'\)), while 'a' and 'b' change states.
   • Therefore, this group simplifies to \(c'\).
2. Group 2 (Two 1's):
   • Observe the '1's at positions (1, 0) and (1, 1) in the map (corresponding to minterms m4, m5).
   • These two cells (100, 101) form a group of two.
   • In this group, variable 'a' is always '1' (a), and 'b' is always '0' (\(b'\)), while 'c' changes.
   • Therefore, this group simplifies to \(ab'\).

Deriving the Simplified Boolean Function

The simplified Boolean function is obtained by summing the simplified terms from all the groups. In this case, we have two groups:

Simplified Function \(F(a, b, c) = (\text{term from Group 1}) + (\text{term from Group 2})\)

\(F(a, b, c) = c' + ab'\)

This is the most simplified form of the given Boolean function.