SOP and POS Forms
Notation

SOP Given a Table of Combinations

What is the SOP form for the following 3 input / 1
output digital device?
S
A
B
f
0
0
0
0
0
0
1
0
0
1
0
1
0
1
1
1
1
0
0
0
1
0
1
1
1
1
0
0
1
1
1
1

Computing the SOP (2)

S
A
B
f
minterm name
0
1
0
1
m2
0
1
1
1
m3
1
0
1
1
m5
1
1
1
1
m7
This SOP has 4 minterms:

f = S'AB' + S'AB + SA'B + SAB

Canonical SOP

Boolean functions can use shorthand notation when
in SOP form:

f = S'AB' + S'AB + SA'B + SAB
f(S,A,B) = (m2,m3,m5,m7)
or
f(S,A,B) = m(2,3,5,7)

Canonical SOP Example


f(x1,x2,x3) = m(1,4,5,6)
minterm
x1
x2
x3
f
0
0
0
0
0
1
0
0
1
1
2
0
1
0
0
3
0
1
1
0
4
1
0
0
1
5
1
0
1
1
6
1
1
0
1
7
1
1
1
0
f = x1'x2'x3 + x1x2'x3' + x1x2'x3 + x1x2x3'

Product of Sums Form

An alternate canonical “two-level” format

“Product of sums”  POS

Two levels



OR level followed by AND level
Again, NOT doesn’t count as a level
Not a common as SOP, but can be useful in some
situations

Which ones?

Computing the POS

Identify rows with “0” on output (f = 0)

Represent the input for each 0 row as a maxterm

A logical “sum” of the input bits which guarantees that term
will be “0” (sum of literals)
A
B
f
0
0
0
0
1
1
1
0
0
1
1
0

Canonical POS Example


f(x1,x2,x3) = (M0,M2,M3,M7) = M(0,2,3,7)
maxterm
x1
x2
x3
f
0
0
0
0
0
1
0
0
1
1
2
0
1
0
0
3
0
1
1
0
4
1
0
0
1
5
1
0
1
1
6
1
1
0
1
7
1
1
1
0
f = (x1+x2+x3)(x1+x2'+x3)(x1+x2'+x3')(x1'+x2'+x3')

Example: 3 Way Light Control


L(A,B,C) = m(5,6)
or
L(A,B,C) = M(0,1,2,3,4,7)
SOP:


L = (A B' C)+(A B C')
POS:

L = (A+B+C)(A+B+C')
(A+B'+C)(A+B'+C')
(A'+B+C)(A'+B'+C')
A
B
C
L
0
0
0
0
0
0
1
0
0
1
0
0
0
1
1
0
1
0
0
0
1
0
1
1
1
1
0
1
1
1
1
0

Question:

Under what conditions would POS form be
better?

(assuming we aren’t doing further reductions)

Inverters in Two-Level Circuits

Inverters are not always required for two-level logic

This is why we do not always count them among the cost of
a circuit

Later, we will see that many variables will be
available to us in both normal and inverted form 
don't need to invert them

We show them only for completeness at this point
NAND/NOR Circuits

Completeness of NAND

Any Boolean function can be implemented
using just NAND gates. Why?





Need AND, OR, and NOT
NOT: 1-input NAND (or 2-input NAND with inputs
tied together)
AND: NAND followed by NOT
OR: NAND preceded by NOTs
Likewise for NOR

Using NAND as Universal Logic

NOT

AND

OR

SOP Using NORs & POS Using NANDs

NANDs are natural for SOP networks


You can extend this idea to multi-level circuits as long as
the levels alternate AND/OR/AND/OR ending with OR
You can implement an SOP circuit using only NOR gates


All gates become NORs; just add an extra “inverter” following
the final NOR
NORs are natural for POS networks


You can extend this idea to multi-level circuits as long as
the levels alternate OR/AND/OR/AND ending with AND
You can implement a POS circuit using only NAND gates

All gates become NANDs; just add an extra inverter following
the final NAND

SOP Using NAND Networks

x1
x2
x1
x2
x3
x4
x5
x3
x4
x5
SOP can be implemented
with just NAND gates


“pushing the bubbles”
Every gate just becomes
a NAND!
x1
x2
x3
x4
x5

2x1 MUX Using NANDs

Implement f = S'A + SB with NAND gates only

This one is complicated by the inverter on S!

POS Using NOR Networks

x1
x1
x2
x2
x3
x3
x4
x4
x5
x5
POS can be implemented
with just NOR gates

Every gate just becomes
a NOR
x1
x2
x3
x4
x5

Schematics of DeMorgan’s Laws
(x ∙ y)' = x' + y'
(x + y)' = x' ∙ y'

Universal Logic Families

Any logic function can be designed using only:



AND, OR, NOT
NAND
NOR

These are called “universal logic families”

Actual components are often designed using either
NAND or NOR gates only


NAND and NOR require fewer transistors to build
Just having a single gate design is simpler than having 3!

AND/OR Networks  NAND/NAND

Convert multi-level AND/OR net  NAND/NAND

And Again … But Be Careful
conserve the polarity of
the input/output signals
Some Useful Circuits
Decoders and Multiplexors

Decoder: Popular
combinational logic building
block, in addition to logic gates


0

i0 d1
1 0
i0 d1
0 1
i0 d1
0
0
i1 d2
0 0
i1 d2
0 1
i1 d2
1 1
i1 d2
0
d3
1
d3
0
d3
i1’i0’
i1’i0
AND gate for each output to
detect input combination
n-input decoder:
2n
outputs
0
0 1
So has four outputs, one for
each possible input binary
number
Outputs all 0 if e=0
Regular behavior if e=1
d0
i0 d1
Decoder with enable e


d0
0
0
Internal design


d0
1
0
d3
0
2-input decoder: four possible
input binary numbers


Converts input binary number
to one high output
d0
i1
i0
d0
d1
i1i0’
d2
i1i0
d3
d0
0
1
i0
d1
0
1
i1
d2
0
e d3
1
1
d0
0
1
i0
d1
0
1
i1
d2
0
e d3
0
0
Multiplexor (Mux)

Mux: Another popular combinational building block

Routes one of its N data inputs to its one output, based on binary
value of select inputs




4 input mux  needs 2 select inputs to indicate which input to route
through
8 input mux  3 select inputs
N inputs  log2(N) selects
Like a railyard switch
Mux Internal Design
2×1
i0
d
i1
s0
2×1
i0
i0
d
i1
s0
0
d
1
i1
d
i1
i0 (1*i0=i0)
i0
2×1
0
s0
1
0
a
2x1 mux
0 s0
4 1
i0
i1
i2
i0
i1
d
d
i2
i3
s1 s0
i3
4x1 mux
s1
s0
i0
(0+i0=i0)
Muxes Commonly Together -- N-bit Mux
2 1
d
s0
a3
b3
i0
i1
a2
b2
2 1
i0
d
i1
s0
2 1
d
s0
a1
b1
i0
i1
a0
b0

i0 2 1
d
i1
s0
A
I0
4-bit
2x1
D
4
B
s0

4
I1
Simplifying
notation:
4
C
4
C
is short
for
s0
c3
s0
c2
c1
c0
Ex: Two 4-bit inputs:


A (a3 a2 a1 a0) and B (b3 b2 b1 b0)
4-bit 2x1 mux (just four 2x1 muxes sharing a select
line) can select between A or B
N-bit Mux Example

Four possible display items



Temperature (T), Average miles-per-gallon (A), Instantaneous mpg (I),
and Miles remaining (M) -- each is 8-bits wide
Choose which to display using two inputs x and y
Use 8-bit 4x1 mux