DLD03

1. Boolean Algebra And
Logic Gates

2. Boolean Algebra Laws

3. Boolean Algebra Laws

4. Proof using Boolean Algebra Laws
Prove: A+A.B =A
=A.(1+B)=A.(1)=A
Prove: A.(A+B) =A
=A.A+A.B=A+A.B=A.(1+B)=A

5. Additional Examples on Using
Boolean Algebra
2. F  A B D  A B C D  A B C D
 A B D  A B D(C  C)
 B D(A  A)
 B D

6. Proof by Truth Table
Any equations can be proved using Truth table (T.T.)
Theorem 6a. A  A B  A

7. Theorem 9a. A  A B  A  B
A B A + B Ā Ā• B A + Ā• B
0 0 0 1 0 0
0 1 1 1 1 1
1 0 1 0 0 1
1 1 1 0 0 1

8. proof of Demorgan theory:A+B=A.B using (T.T.)
A.B
B
A
A+B
A+B
B
A
1
1
1
1
0
0
0
0
0
1
0
1
1
0
0
1
0
0
1
0
1
0
0
0
0
1
1
1

9. NAND and NOR are universal gates
Any function can be implemented using only NAND
or only NOR gates. How can we prove this?
(Proof for NAND gates)
Any boolean function can be implemented using AND, OR
and NOT gates.So if AND, OR and NOT gates can be
implemented using NAND gates only, then we prove our
point.

10. 1. Implement NOT using NAND
2. Implementation of AND using NAND
A.B=A.B

11. A B  A  B
3. Implementation of OR using NAND

12. (Proof for NOR gates)
Any boolean function can be implemented using AND, OR
and NOT gates.So if AND, OR and NOT gates can be
implemented using NOR gates only, then we prove our
point.
A A
1. Implement NOT using NOR

13. A B  A  B
3. Implementation of OR using NAND

14. 2. Implementation of AND using NOR
A B  A  B
3. Implementation of OR using NOR
A  B  A  B

15. Designing Logic Circuits Using NAND Gates only
F  AB  CD
 AB  CD

16. F  ABC  D
 ABC  D

17. Designing Logic Circuits Using NOR Gates only
(A B)(AC) A B AC

18. Standard Forms For Logic Functions
Two standard forms of Boolean
expressions:
• Sum of Products.
• Product of Sums.

19. Sum of Products
• The logic function is written as a simple sum of terms is
called a sum-of-products form.
• In each term, variables are connected with AND
operators.
Example:
1
1
1
1
7
1
0
1
1
6
0
1
0
1
5
1
0
0
1
4
1
1
1
0
3
1
0
1
0
2
0
1
0
0
1
0
0
0
0
0
L
Z
Y
X
Row
L  X Y Z  X Y Z  X Y Z  X Y Z  X Y Z

20. • For each individual product term with the output logic is “1”,
it is called a minterm.
• The logic equation in the previous example can be written as:


m
7)
6,
4,
3,
(2,
L
Row numbers (decimal
equivalent)
L=m2+m3+m4+m6+m7
m=minterm (standard product)

21. Product of Sums
• Product-of-sums form is written as a simple product of terms.
• The OR operators are interconnected variables in each term.
Example:
1
1
1
1
7
1
0
1
1
6
0
1
0
1
5
1
0
0
1
4
1
1
1
0
3
1
0
1
0
2
0
1
0
0
1
0
0
0
0
0
L
Z
Y
X
Row
L  (X Y  Z) (X Y  Z) (X Y  Z)
L  X Y  Z
L  X Y  Z

22. • For each individual summing term with the output logic is “0”, it is
called a maxterm.
• The logic equation can be expressed as
L  
M
(0,1,5)
L=(M0).(M1).(M5)
M=Maxterm (standard sum)

23. EX: Design the following logic circuit according to the
shown truth table:
A B C F
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 0
1 1 1 1
0
1
2
3
4
5
6
7
Decimal

24. H.W
Design a digital logic circuit that will activate an alarm if a door
or window is open during non-business hours. Assuming that the
status of the clock, door, window are monitored using sensors that
produce digital output by the variables C, D and W.
Clock C = 0 (non-business hours) 1 business hours)
Door D = 0 (closed) 1 opened)
Window W = 0 (closed) 1 opened)
Alarm A = 0 (off) 1 (on)

25. c d w A
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 1
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 0
0
1
2
3
4
5
6
7
Decimal
A=Σm (1,2,3)
A=m1+m2+m3
A = c d w + c d w + c d w
By using Boolean Algebra Laws:
A = c d w + c d (w + w )
A = c d w + c d
A = c ( d w + d )
A = c ( w + d )
A = c w + c d
Logic
circuit