2. •Further Boolean algebra, de Morgan’s
theorem
•Karnaugh maps with up to four variables
•Simplification and optimisation of circuits
using these techniques
3. 3
Simplification of Logic Circuits using
Boolean Algebra
• Need to apply the laws, rules and theorems
of Boolean Algebra to simplify Boolean
Expressions
• Simplification means fewer gates for the
same logic functions
• If equivalent logic function may be achieved
with fewer components, the result will be
increased reliability and decreased cost of
manufacture
4. 5
Rules of Boolean Algebra
AND Operator OR Operator
A.0 ≡ 0 A + 0 ≡ A
A.1 ≡ A A + 1 ≡ 1
A.A ≡ A A + A ≡ A
Note that the rules for OR operator can be obtained from AND operator
by changing the operator, and inverting the logic values (vice versa)
=> Duality of Boolean Algebra
1AA =+0A.A =
6. 9
DeMorgan’s Theorem 1
Input
x y
0 0
0 1
1 0
1 1
0
1
1
1
1
0
0
0
1
1
0
0
1
0
1
0
1
0
0
0
Notice the third and the last column are the same
verifying DeMorgan’s Theorem 1
Y.XYX =+
YX + YX + X Y Y.X
7. 10
DeMorgan’s Theorem 2
Input
x y X.Y
0 0
0 1
1 0
1
0
0
0
1
1
1
1
0
1
1
0
0
1
0
1
0
1
1
1
0
Notice the third and the last column are the
same verifying DeMorgan’s Theorem 2
YXY.X +=
Y.X X Y
10. Implementation Example:
Using Basic Logic Gates
• X = A.B + C.D
• A straight-forward method would be to use two
AND gates from a 7408 IC and one OR gate
from a 7432 IC
11. Implementation Example:
Using only NAND gates
redundant
This solution only
uses three NAND
gates from one 7400
IC
=> Saves PCB area,
component costs and
power consumption!!
12. In an alarm system, the alarm will sound off
(Output X = 1) as long as security areas “B”
and “C” are breached. Inputs A, B, C
represents the 3 security areas, with logic 1
representing that the security is breached.
1)Draw the truth table for this logic function
with 3 inputs and 1 output.
2) Design the logic circuit
13. Inputs Output
A B C X
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 1
Inputs Output
A B C X
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 1
Truth Table for the alarm system
14. Design of Alarm system
Canonical/Standard SOP Expression:
Canonical/Standard POS Expression:
It can be
shown using
Boolean
Algebra that
these 2
expressions
are equivalent.
Try it
yourself !!
Inputs Output
Minterm Maxterm
A B C X
0 0 0 0 A+B+C
0 0 1 0 A+B+C’
0 1 0 0 A+B’+C
0 1 1 1 A’.B.C
1 0 0 0 A’+B+C
1 0 1 1 A.B’.C
1 1 0 1 A.B.C’
1 1 1 1 A.B.C
Alarm will sound off (Output X = 1) when at least 2 out of 3 security areas are
breached.
C.B.AC.B.AC.B.AC.B.AX +++=
).CBA).(CBA).(CBA).(CBA(X ++++++++=
15. Design of Alarm system
Simplified SOP expression:
Universal of NAND gates to implement the alarm circuit.
C.AC.BB.AX ++=
Universal of NAND gates