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 closed closed open open
Q closed open closed open
State on off off off
– closed/on => T
– open/off => F
T
F
P
T
F
F
T
Q
T
F
F
F
State
T
F

Switching Table
• Switches in parallel
P closed closed open open
Q closed open closed open
State on on on off
– closed/on => T
– open/off => F
T
F
P
T
F
F
T
Q
T
F
T
T
State
T
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
0
1
1
P
0
1
0
1
Q
0
Output
X
X
X
R
X

Example
P v Q
~(P ^ Q)
P ^ Q
P
0
0
1
1
Input
Q
0
1
0
1
(P v Q) ^ ~(P ^ Q)
Output
R
0
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 |)