Download Logic Gates, Truth Tables, and Realization of Boolean Formulas and more Study notes Logic in PDF only on Docsity!
1
Lecture 4
Logic gates and truth tables
Implementing logic functions
Canonical forms
Sum-of-products
Product-of-sums
2
Logic gates and truth tables
AND X•Y XY
OR X + Y
NOT X' X
X Y Z
X Y Z 0 0 0 0 1 0 1 0 0 1 1 1
X Y
Z
X (^) Y
_
X Y Z 0 0 0 0 1 1 1 0 1 1 1 1
X Y 0 1 1 0
3
Logic gates and truth tables
NAND
NOR
X • Y XY X^ Y^ Z
0 0 1 0 1 1 1 0 1 1 1 0
X Y Z 0 0 1 0 1 0 1 0 0 1 1 0
X +Y
X Y Z
Z
X Y
4
Logic gates and truth tables
XOR
XNOR
X Y Z
Z
X Y
X Y Z 0 0 0 0 1 1 1 0 1 1 1 0
X Y Z 0 0 1 0 1 0 1 0 0 1 1 1
X ⊕Y
X ⊕Y
Realizing Boolean formulas
F = (A•B)’ + C•D F = C•(A+B)’
F
A B
C D
A B C
F
Realizing truth tables
Given a truth table
1. Write the Boolean expression
2. Minimize the Boolean expression
3. Draw as gates
7
Example
A B C F
F = A’BC’+A’BC+AB’C+ABC
= A’B(C’+C)+AC(B’+B)
= A’B+AC
8
Example: Binary full adder
1-bit binary adder
Inputs: A, B, Carry-in
Outputs: Sum, Carry-out
A
B
Cin Cout
Sum Adder
A B Cin Cout Sum 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1
Cout = A'BCin + AB'Cin + ABCin' + ABCin
Both Sum and Cout can be minimized.
Sum = A'B'Cin + A'BCin' + AB'Cin' + ABCin
9
Full adder: Sum
Before Boolean minimization Sum = A'B'Cin + A'BCin'
After Boolean minimization Sum = (A⊕B) ⊕ Cin
10
Full adder: Carry-out
Before Boolean minimization Cout = A'BCin + AB'Cin
After Boolean minimization Cout = BCin + ACin + AB
Preview: 2-bit ripple-carry adder
A 1 B 1
Cin Cout
Sum 1
A 2 B 2
Sum 2
0 Cin Cout
Overflow
A
Sum
Cin Cout
B
1-Bit Adder
Many possible mappings
Many ways to map expressions to
gates
Example: Z = A•B•(C+D) = A•(B•(C+D))
_ _ _ _
19
Maxterms
Variables appear exactly once in each
maxterm in true or inverted form (but
not both)
A B C maxterms 0 0 0 A+B+C M 0 0 1 A+B+C' M 0 1 0 A+B'+C M 0 1 1 A+B'+C' M 1 0 0 A'+B+C M 1 0 1 A'+B+C' M 1 1 0 A'+B'+C M 1 1 1 A'+B'+C' M
short-hand notation
F in canonical form: F(A,B,C) = ΠM(0,2,4) = M0 • M2 • M = (A+B+C)(A+B'+C)(A'+B+C)
20
Example: F = AB+C
21
From SOP to POS and back
Minterm to maxterm
Use maxterms that aren’t in minterm
expansion
F(A,B,C) = ∑m(1,3,5,6,7) = ∏M(0,2,4)
Maxterm to minterm
Use minterms that aren’t in maxterm
expansion
F(A,B,C) = ∏M(0,2,4) = ∑m(1,3,5,6,7)
22
From SOP to POS and back
Minterm of F to minterm of F'
Use minterms that don’t appear
F(A,B,C) = ∑m(1,3,5,6,7) F' = ∑m(0,2,4)
Maxterm of F to maxterm of F'
Use maxterms that don’t appear
F(A,B,C) = ∏M(0,2,4) F' = ∏M(1,3,5,6,7)
SOP, POS, and DeMorgan's
Sum-of-products
F' = A'B'C' + A'BC' + AB'C'
Apply DeMorgan's to get POS
(F')' = (A'B'C' + A'BC' + AB'C')'
F = (A+B+C)(A+B'+C)(A'+B+C)
SOP, POS, and DeMorgan's
Product-of-sums
F' = (A+B+C')(A+B'+C')(A'+B+C')(A'+B'+C')
Apply DeMorgan's to get SOP
(F')' = ((A+B+C')(A+B'+C')(A'+B+C')(A'+B'+C'))'
F = A'B'C + A'BC + AB'C + ABC
