Chapter 1
The Logic of Compound Statements
Section 1.4
Digital Logic Circuits
Digital Circuits
• Electrical circuits can be fashioned to mimic
logic tables.
• Types of switches:
– open
– closed
• Types of circuits:
– series
– parallel
Switching Table
• Switches in series
P
Q
State
closed
closed
on
closed
open
off
open
closed
off
open
open
off
– closed/on => T
– open/off => F
P
Q
State
T
T
T
T
F
F
F
T
F
F
F
F
Switching Table
• Switches in parallel
P
Q
State
closed
closed
on
closed
open
on
open
closed
on
open
open
off
– closed/on => T
– open/off => F
P
Q
State
T
T
T
T
F
T
F
T
T
F
F
F
Basic Digital Logic Gates
Combinational Circuits
• Combinational circuits are composed of one or more basic
gates where the output of the circuit is based on the input
at that instant in time.
• Rules of Combinational Circuits
– Never combine two input wires.
– A single input wire can be split and used as input for two
separate gates.
– An output wire can be used as input.
– No output of a gate can feedback into that gate.
• Sequential circuits are circuits that include feedback. Their
output depends on previous input. These circuits are used
to build circuits that can remember (memory circuits).
Example
Input-Output Table
• Input-output table is a truth table for a
combinational circuit. It shows the output of
the circuit given a set of inputs.
Input
Output
P
Q
R
0
0
X
0
1
X
1
0
X
1
1
X
Example
PvQ
(P v Q) ^ ~(P ^ Q)
P^Q
~(P ^ Q)
Input
Output
P
Q
R
0
0
0
0
1
1
1
0
1
1
1
0
Boolean
• A combinational circuit can be expressed as a
Boolean expression.
• George Boolean was an English
mathematician who founded symbolic logic.
• Boolean variable is a variable that has only
two possible values (T/F, on/off, 1/0).
• Boolean expression is composed of Boolean
variables and connectives (~, v, ^ )
Boolean Expression Circuits
• A Boolean expression can be converted to a
combinational digital logic circuit by using the
Boolean variables as inputs and matching the
connectives (~, v, ^) with their gate equivalent
(NOT, OR, AND).
• Example
– (~P ^ Q) v ~Q
Circuit from I/O Table
• A circuit can be constructed from any I/O
table.
• A circuit constructed in this form will be
composed of a set of AND gates connected by
OR gates. R^S v ~R^S v R^~S
Example
1^1^1 v 1^0^1 v 1^0^0
P^Q^R v P^~Q^R v P^~Q^~R
Equivalent Circuits
• Two circuits are equivalent if there I/O tables
are equivalent.
• As with logic expressions, digital circuits may
be simplified through logic theorem 1.1.1, aka
Boolean Algebra.
Example
• ((P ^ ~Q) V (P ^ Q)) ^ Q
– (P ^ (~Q V Q)) ^ Q (distributive)
– (P ^ (Q v ~Q)) ^ Q (commutative)
– (P ^ t) ^ Q (negation)
– P ^ Q (identity)
• Inspection of the I/O table reveals the
simplified circuit.
NAND and NOR Gates
• NAND or NOR gates can be used to simplify a circuit as they
are primitive gates, i.e. all gates can be built from them. (NOT,
AND, OR, XOR, etc.)
NAND and NOR
• NAND
– logic symbol is (Sheffer Stroke) |
– P|Q  ~(P ^ Q)
• NOR
– logic symbol is (Peirce Arrow) 
– PQ  ~(P v Q)
NAND (Sheffer Stroke) Example
• Show that the Sheffer Stroke (NAND) can be
used to implement ~ (NOT)
– ~P  P | P
– ~P  ~(P ^ P) (idempotent)
–  P | P (definition of |)