Logic Gates, Truth Tables, and Realization of Boolean Formulas, Study notes of Logic

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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⊕ 5 Realizing Boolean formulas
F = (A•B)’ + C•D
F = C•(A+B)’ F A B C D A B C F 6 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 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 0 1 1 1 1 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 0 1 1 0 1 0 0 1 0 0 0 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' + AB'Cin' + ABCin After Boolean minimization Sum = (A⊕B) ⊕ Cin 10 Full adder: Carry-out Before Boolean minimization Cout = A'BCin + AB'Cin + ABCin' + ABCin After Boolean minimization Cout = BCin + ACin + AB 11 Preview: 2-bit ripple-carry adder A1 B1 CoutCin Sum1 A2 B2 Sum2 CoutCin0 Overflow A Sum CoutCin B 1-Bit Adder 12 Many possible mappings
Many ways to map expressions to gates
Example: Z = A•B•(C+D) = A•(B•(C+D)) _ _ _ _