Digital Logic
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Boolean Logic and Logic Gates: From Boolean Algebra to Digital Circuits, Slides of Computer Science
An introduction to digital logic, focusing on boolean algebra and its application to logic gates. Topics include logic circuits built from components called logic gates, which correspond to boolean operations +, *, and not. And, or, not, nand, nor, xor, and xnor gates, their properties, and their use as universal logic gates. The document also includes examples and exercises.
Typology: Slides
2012/2013
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Digital Logic
Boolean Algebra to Logic Gates
- Logic circuits are built from components called
logic gates.
- The logic gates correspond to Boolean operations
- Binary operations have two inputs, unary has one
OR
AND
NOT
A
B
A+B
Logic Gate:
Parallel Circuit:
A
B
A B A+B
A+B Truth Table:
OR
NOT
A
A’ or A
Logic Gate: (also called an inverter)
Single-throw Double-pole Switch:
A
a A 0 1 1 0
Truth Table:
A’ or A
n -bit Inputs
- For convenience, it is sometimes useful to think
of the logic gates processing n -bits at a time. This
really refers to n instances of the logic gate, not a
single logic date with n -inputs.
Logic Circuits ≡ Boolean Expressions
- All logic circuits are equivalent to Boolean expressions and any boolean expression can be rendered as a logic circuit.
- AND-OR logic circuits are equivalent to sum-of-products form.
- Consider the following circuits:
A
C
B abc
aBc
Ab y=abc+aBc+Ab
y
A
B
C
Y
y=aB+Bc
XOR and XNOR Gates
• XOR is used to choose between two mutually
exclusive inputs. Unlike OR, XOR is true only
when one input or the other is true, not both.
XOR
XNOR
A B AB 0 0 0 0 1 1 1 0 1 1 1 0 A B A B 0 0 1 0 1 0 1 0 0 1 1 1
Properties of NAND AND NOR
• NAND and NOR have special properties, but
neither satisfies the distributive or associative
laws.
NAND NOR
x1=X x0=X x 0=1 x 1= x x=X x x=X x y=X+Y x y=XY X Y=x+y X Y=xy not (x y)=xy not (x y)=x+y
It should be clear by looking at these properties that NAND and NOR are duals.
NAND and NOR as Universal Logic Gates (cont)
- Here are the NOR
equivalents:
- NAND and NOR
can be used to reduce the number of required gates in a circuit.
Example Problem
• A hall light is controlled by two light switches, one
at each end. Find (a) a truth function, (b) a
Boolean expression, and (c) a logic network that
allows the light to be switched on or off by either
switch.
x y f(x,y) 0 0 0 0 1 1 1 0 1 1 1 0
(What kind of gate has this truth table?