Questions: Simplification: F(A, B, C, D) = sum(0,1,4,6,7,12,13,14,15) A B / C D 00 01 11 10 00 1 1 0 0 01 1 0 1 1 11 1 1 1 1 10 0 0 0 0 Give me the simplified circuit based on k-map

Transcript text: Simplification: \[ F(A, B, C, D)=\sum(0,1,4,6,7,12,13,14,15) \] \begin{tabular}{l|l|l|l|l|} $A B / C D$ & 00 & 01 & 11 & 10 \\ \hline 00 & 1 & 1 & 0 & 0 \\ 01 & 1 & 0 & 1 & 1 \\ 11 & 1 & 1 & 1 & 1 \\ 10 & 0 & 0 & 0 & 0 \end{tabular} Give me the simplified circuit based on k-map

Solution

Solution Steps

To simplify the given Boolean function using a Karnaugh map (K-map), we will first plot the given minterms on a 4-variable K-map. Then, we will group the 1s into the largest possible power-of-two rectangles to find the simplified expression. Each group will correspond to a product term in the simplified Boolean expression.

Step 1: Define the Minterms

The given function $ F(A, B, C, D) $ is defined by the minterms $ \{0, 1, 4, 6, 7, 12, 13, 14, 15\} $. These minterms correspond to the binary representations of the inputs where the function evaluates to 1.

Step 2: Grouping in K-map

Using a Karnaugh map, we identify groups of 1s. The groups formed based on the minterms are:

Group 1: $ \{0, 1, 4, 5\} $ corresponds to $ \sim A \cdot \sim B $
Group 2: $ \{4, 6, 12, 14\} $ corresponds to $ \sim C \cdot \sim D $
Group 3: $ \{6, 7, 14, 15\} $ corresponds to $ B \cdot C $
Group 4: $ \{12, 13, 14, 15\} $ corresponds to $ A \cdot B $

Step 3: Forming the Simplified Expression

Combining the product terms from the groups, we obtain the simplified Boolean expression: \[ F(A, B, C, D) = (A \cdot B) + (B \cdot C) + (\sim A \cdot \sim B) + (\sim C \cdot \sim D) \]

Final Answer

The simplified expression for the function $ F(A, B, C, D) $ is: \[ \boxed{F(A, B, C, D) = (A \cdot B) + (B \cdot C) + (\sim A \cdot \sim B) + (\sim C \cdot \sim D)} \]

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