DHARM
N-COM\APPENDIX.PM5
BOOLEAN ALGEBRA 373
That can be represented by the K-map shown in Fig. A.22.
pq
rs
pq′′
pq′
pq
pq′
rs′ rs′ rsrs′
11
1
111
1
00
0
0
000
0
0
Fig. A.22
We can group the cells recognized by 1’s as,
- Top two cells of 1’s can grouped to the lowest two cells of 1’s; return the expression
p′q′ r′ + p q′ r′ = (p′ + p) q′ r′ = 1. q′ r′ = q′ r′;
- Combing four 1’s at corner gives the simplified expression, q′ s′
- Grouping of top two 1’s at second column gives the expression, p′ r′ s
Therefore simplified Boolean function is,
F (p, q, r, s) = q′ r′ + q′ s′ + p′ r′ s (SoP)
To obtain the product of sum expression we shall combine the cells marked with 0’s.
Since, combining the cells of 0’s represent the minterms not included in the function, hence it
denotes the complement of F that gives the simplified function in PoS.
- Combing 0’s that lies middle rows and two end columns gives the expression,
q r′ s′ + q r s′ = q s′
- Combing all 0’s of third row gives the expression, p q
- Combining all 0’s of third column gives the expression, r s (Fig. A.23)
pq
rs
pq′′
pq′
pq
pq′
rs′ rs′ rsrs′
1
1
10
0
0000
0 0
0
1
11
1
Fig. A.23
