1. The Logic of Compound Statements
Chapter 1
The Logic of Compound Statements

2. Digital Logic Circuits
Section 1.4
Digital Logic Circuits

3. Digital Circuits
Electrical circuits can be fashioned to mimic logic tables.
Types of switches:
open
closed
Types of circuits:
series
parallel

4. Switching Table Switches in series closed/on => T open/off => FQ
State
closed on
open
off
Series switch mimics an AND table P Q
State T F

5. Switching Table Switches in parallel closed/on => T
open/off => F P Q
State
closed on
open
off
Series switch mimics an OR table P Q
State T F

6. Basic Digital Logic Gates

7. 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).

8. Example Trace the propagation of the inputs through the circuit.
input: P=1 Q=0, output: R = 0
input: P=0, Q=1, R=1, output: S =0

9. 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 X 1

10. Example P v Q (P v Q) ^ ~(P ^ Q) ~(P ^ Q) P ^ Q Input Output P Q R 11
This circuit is equivalent to XOR. That is it is 1 (TRUE) only when the inputs are opposite and 0 (FALSE) when the inputs have the same value.

11. 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, ^ )

12. 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

13. 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

14. Example
1^1^1 v 1^0^1 v 1^0^0
P^Q^R v P^~Q^R v P^~Q^~R

15. 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.

16. 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.

17. 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.)

18. NAND and NOR NAND NOR logic symbol is (Sheffer Stroke) |
P|Q  ~(P ^ Q)
NOR
logic symbol is (Peirce Arrow) 
PQ  ~(P v Q)

19. 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 |)