Given the Boolean function . Reduce the above expression by using 4-variable Karnaugh map, showing…
Computer Science20175 marksDrawing
Given the Boolean function $F(P, Q, R, S) = \pi(0, 1, 2, 4, 5, 6, 8, 10)$.
(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
$F(P,Q,R,S) = \pi(0,1,2,4,5,6,8,10)$. Plotting the 0s on a 4-variable K-map (rows PQ, columns RS in the order 00, 01, 11, 10):
PQ \ RS
00
01
11
10
00
0
0
1
0
01
0
0
1
0
11
1
1
1
1
10
0
1
1
0
Groups of 0s: there is no octet and no pair is needed.
Quad $\{M_0, M_1, M_4, M_5\}$: $(P + R)$
Quad $\{M_0, M_2, M_4, M_6\}$: $(P + S)$
Quad $\{M_0, M_2, M_8, M_{10}\}$ (four corners): $(Q + S)$
Reduced expression: $F = (P + R) \cdot (P + S) \cdot (Q + S)$ (verified with the boolean tool).
Answer (ii)
AI
Logic gate diagram for $F = (P + R)(P + S)(Q + S)$: three 2-input OR gates with inputs (P, R), (P, S) and (Q, S) feed one 3-input AND gate whose output is F.
