Standard (Canonical) Forms

• Canonical SOP:
  \(f(A,B,C,D,E)=\Sigma m(2,3,5,11,13,19,21,27,28,29)\)

Full Truth Table (0–31)

(A as MSB, E as LSB)

idx	A B C D E	f	idx	A B C D E	f
0	0 0 0 0 0	0	16	1 0 0 0 0	0
1	0 0 0 0 1	0	17	1 0 0 0 1	0
2	0 0 0 1 0	1	18	1 0 0 1 0	0
3	0 0 0 1 1	1	19	1 0 0 1 1	1
4	0 0 1 0 0	0	20	1 0 1 0 0	0
5	0 0 1 0 1	1	21	1 0 1 0 1	1
6	0 0 1 1 0	0	22	1 0 1 1 0	0
7	0 0 1 1 1	0	23	1 0 1 1 1	0
8	0 1 0 0 0	0	24	1 1 0 0 0	0
9	0 1 0 0 1	0	25	1 1 0 0 1	0
10	0 1 0 1 0	0	26	1 1 0 1 0	0
11	0 1 0 1 1	1	27	1 1 0 1 1	1
12	0 1 1 0 0	0	28	1 1 1 0 0	1
13	0 1 1 0 1	1	29	1 1 1 0 1	1
14	0 1 1 1 0	0	30	1 1 1 1 0	0
15	0 1 1 1 1	0	31	1 1 1 1 1	0
# Minterm Table (index → A B C D E)

 2 → 0 0 0 1 0
 3 → 0 0 0 1 1
 5 → 0 0 1 0 1
11 → 0 1 0 1 1
13 → 0 1 1 0 1
19 → 1 0 0 1 1
21 → 1 0 1 0 1
27 → 1 1 0 1 1
28 → 1 1 1 0 0
29 → 1 1 1 0 1

5-Variable K-map (Gray code order)

I’ll split by \(A\): two 4×4 maps with rows \(BC=00,01,11,10\) and columns \(DE=00,01,11,10\).

A = 0

          DE→   00  01  11  10
         +----------------------
BC = 00  |  0   0   1   1
BC = 01  |  0   1   0   0
BC = 11  |  0   1   0   0
BC = 10  |  0   0   1   0

A = 1

          DE→   00  01  11  10
         +----------------------
BC = 00  |  0   0   1   0
BC = 01  |  0   1   0   0
BC = 11  |  1   1   0   0
BC = 10  |  0   0   1   0

Minimized SOP

\[ \boxed{\,f = C\overline{D}E \;+\; \overline{C}DE \;+\; ABC\overline{D} \;+\; \overline{A}\,\overline{B}\,\overline{C}D\,} \]