Boolean Function

1. Boolean Function
Dr. (Mrs.) Gargi Khanna
Associate Professor
Electronics & Communication Engineering Department
National Institute of Technology, Hamirpur

2. Boolean function
• A Boolean function has:
• At least one Boolean variable,
• At least one Boolean operator, and
• At least one input from the set {0,1}.
• It produces an output that is also a member of the
set {0,1}.

3. Truth Table
• Addition of 2 bits:
A B Sum Carry
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1
Sum= 𝐴′
𝐵 + 𝐴𝐵′ 𝐶𝑎𝑟𝑟𝑦 = 𝐴𝐵
Sum= 𝐴 + 𝐵 . (𝐴′
+ 𝐵′) 𝐶𝑎𝑟𝑟𝑦 = 𝐴 + 𝐵 . 𝐴 + 𝐵′
. (𝐴′
+ 𝐵)

4. Truth Table
• Subtraction of 2 bits:
A B Difference Borrow
0 0 0 0
0 1 1 1
1 0 1 0
1 1 0 0
Difference= 𝐴′
𝐵 + 𝐴𝐵′ 𝐵𝑜𝑟𝑟𝑜𝑤 = 𝐴′𝐵
Difference = 𝐴 + 𝐵 . (𝐴′ + 𝐵′) 𝐵𝑜𝑟𝑟𝑜𝑤 = 𝐴 + 𝐵 . 𝐴′ + 𝐵 . (A’ +B’)

5. Truth Table
• Addition of 3 bits:
A B C Sum Carry
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 1 1 0 1
1 0 0 1 0
1 0 1 0 1
1 1 0 0 1
1 1 1 1 1
Sum= 𝐴′ 𝐵′ 𝐶 + 𝐴′ 𝐵𝐶′ + 𝐴𝐵′ 𝐶′ + 𝐴𝐵𝐶
𝐶𝑎𝑟𝑟𝑦 = 𝐴′
𝐵𝐶 + 𝐴𝐵′
𝐶 + 𝐴𝐵𝐶′
+ABC
Sum= (𝐴 + 𝐵 + 𝐶). (𝐴 + 𝐵′+𝐶′)(𝐴′ + 𝐵 + 𝐶′)(𝐴′ + 𝐵′ + 𝐶)

6. Two-Level Implementations
• Boolean functions in either SOP or POS forms
can be implemented using 2-Level
implementations.
• For SOP forms AND gates will be in the first level
and a single OR gate will be in the second level.
• For POS forms OR gates will be in the first level
and a single AND gate will be in the second
level.
• SOP forms can be implemented using only
NAND gates, while POS forms can be
implemented using only NOR gates.

8. F = XZ + Y’Z + X’YZ
XZ
Y’Z
X’YZ
XZ . Y’Z . X’YZ = XZ + Y’Z + X’YZ
F = XZ + Y’Z + X’YZ

9. Implement the following POS function
F = (X+Z) (Y’+Z) (X’+Y+Z)

10. Standard SOP
• Write minterms of BC + A.
Solution : BC + A Write terms
• XBC + AXX Add Xs where letters are missing
• A’BC, ABC Vary Xs in XBC
• AB’C’, AB’C, ABC’, ABC Vary Xs in AXX
• Thus BC + A = A’ BC +ABC+ AB’C’ + AB’C + A B
C’ + A B C
• or BC + A = A’BC + ABC + AB’C’ + AB’C + ABC’

11. Standard SOP
F(A.B,C) = Σm(3, 7, 4, 5, 6)

12. Complementary Nature of Min & Max terms

13. F=Σm( 15, 11, 14, 13, 12)
F=П M (0,1,2,3,4,5,6,7,8,9,10)

14. Standard POS
Write full form of the expression Y(A,B,C) = П M (0, 1, 3, 4)

15. Standard SOP & POS
• F= B’+A’C
• X0X, 0X1
• (000, 001,100,101), (001,011)
• F= Ʃm(0,1,4,5,3,8)
F(A,B,C)=(A+B)(A’+C)
• (0+0+X)
(1+X+0)=(0+0+0),(0+0+1),(1+0+0),(1+1+0)
• F=ПM(1,0,4,6)

16. Happy Learning