Simplification of switching functions
• Simplify – why?
– Switching functions map to switching circuits
– Simpler function  simpler circuit
– Reduce hardware complexity
– Reduce size and increase speed by reducing
number of gates
• Simplify – how?
– Using the postulates
– Ad-hoc
Simplification of switching functions
• Simplify – what?
– SOP/POS form has products/sums and literals
• Literal: each appearance of a variable or its
complement
– Minimize number of sums/products
• Reduces total gate count
– Minimize number of variables in each sum/product
• Reduces number of inputs to each gate
• PLDs have fixed # of inputs; only the number of terms need
to be minimized there
Simplification of switching functions
Simplification using postulates
Simplification using Karnaugh maps
Karnaugh maps
• Karnaugh map (also K-map) is a graphic tool,
pictorial representation of truth table
– Extension of the concepts of truth table, Venn
diagram, minterm
– Transition from Venn diagram to minterm
Karnaugh maps
– Adjacencies are preserved when going from c) to d)
• They are the same, only the areas are made equal in d),
which preserves adjacencies
• Subscripts are dropped in e); realize that 2&3 is A; 1&3 is B
• In f) the labels change and become 0 and 1
– Each square of the K-map is 1 row of the TT
Karnaugh maps
• Might start with rectangles initially and get the
same result
 A
B
– Each square of the K-map is 1 row of the TT
Karnaugh maps
• One to one correspondence between K-map
squares and maxterms
A
A+B  M0 = m0 = AB
B
A
A+B  M3 = m3 = AB
B
Karnaugh maps
• One to one correspondence between K-map
squares and maxterms
A
A+B  M2 = m2 = AB
B
A
A+B  M1 = m1 = AB
B
3-variable K-maps
3-variable K-maps
• Constructing 3-variable K-maps
A
A
B
0 1
1 0
0
flip 
0
1
1
C=0
C=1
abutt
CA
B
00 01 11 10
0
1
B
3-variable K-maps
• Constructing 3-variable K-maps
A
A
B
0 1
CB 1 0
0
C=0
00
1
01
C=0
11
A
10
B
0 1
1
C=1
0
4-variable K-maps
5-variable K-maps
5-variable K-maps
6-variable K-maps
6-variable K-maps
Plotting functions in canonical form
Plotting functions in canonical form