Given the Boolean function . Reduce the above expression by using 4-variable Karnaugh map, showing…
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Given the Boolean function $F(A, B, C, D) = \Sigma (2, 3, 4, 5, 6, 7, 8, 10, 11)$.
(i)[4.0]
Reduce the above expression by using 4-variable Karnaugh map, showing the various groups (i.e. octal, quads and pairs).
(ii)[1.0]
Draw the logic gate diagram for the reduced expression. Assume that the variables and their complements are available as inputs.
Draw: Logic gate diagram for the reduced expression
Answer
Answer (i)
AI
$F(A,B,C,D) = \Sigma(2,3,4,5,6,7,8,10,11)$. Plotting the 1s on a 4-variable K-map (rows AB, columns CD in the order 00, 01, 11, 10):
AB \ CD
00
01
11
10
00
0
0
1
1
01
1
1
1
1
11
0
0
0
0
10
1
0
1
1
Groups: there is no octet.
Quad $\{m_4, m_5, m_6, m_7\}$: $A'B$
Quad $\{m_2, m_3, m_{10}, m_{11}\}$ (wraps from the top row to the bottom row): $B'C$
Pair $\{m_8, m_{10}\}$: $AB'D'$
Reduced expression: $F = A'B + B'C + AB'D'$ (verified with the boolean tool).
Answer (ii)
AI
Logic gate diagram for $F = A'B + B'C + AB'D'$: three AND gates with inputs (A', B), (B', C) and (A, B', D') feed one 3-input OR gate whose output is F. The complemented inputs A', B', D' are taken directly as inputs.
