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CPSC 171 Introduction to Computer Science
Boolean Logic, Gates, & Circuits
Announcements
Read Chapter 4
Exam, Oct 2nd in class
Boolean Logic
A Boolean variable, A, is either true or falseA Boolean expression, (A AND B), evaluates to either true or falseBoolean operators include: AND (& • ) OR ( + ) NOT (a bar ' ¬ ~)
Boolean Operators
a AND btrue only when A and B are both true
a OR btrue when A is true, B is true, or both are
true
NOT atrue when A is false
Truth Tables
Truth tables can be used to capture when an expression is true, given its inputs
a b a AND b
0 0 1
0 1 1
1 0 1
1 1 1
You make truth tables for AND and NOT
Example Boolean Expressions
(a AND b) OR (NOT a AND c)a·b + ~a·cab+āc
Truth tables can be made for complex expressions as well
a b Value
0 0 1
0 1 0
1 0 0
1 1 1
Example: (a AND b) OR ((NOT b) and (NOT a))
Boolean Logic (continued)
Gates
Gates Hardware devices built from transistors
to mimic Boolean logic An electronic device that operates on a
collection of binary inputs to produce a single binary output
AND gate (page 161 in text) Two input lines, one output line Outputs a 1 when both inputs are 1
Gates (continued)
OR gate (page 163 in text) Two input lines, one output line Outputs a 1 when either input is 1
NOT gate (page 161 in text One input line, one output line Outputs a 1 when input is 0 and vice
versa
Figure 4.15The Three Basic Gates and Their Symbols
Circuits
A collection of logic gates that transforms a set of binary inputs into a set of binary outputsWire gates together keeping constraints for the number of inputs to any gate
Example Circuit
If a, b, c, and d are all true the output can be determined by tracing through the circuit
a b
c d
output
1
1
1
1
1
1
0
0
Designing CircuitsA circuit construction algorithm
1. Truth Table ConstructionDetermine outputs for every possible input
2. Sub-expression Construction (using AND and NOT gates)For each output find the rows that are 1 and build a
sub-expression that is true for the exact input
3. Sub-expression combination (using OR gates)Take each subexpression and combine them, 2 at a
time, using OR gates
4. Circuit Diagram ProductionConstruct final circuit by converting Boolean operators
into gates
Example Circuit Design
Design a 3-input circuit that is true if exactly two inputs are true, and false otherwise
You Try it: Design a 2-input circuit that is true if the inputs are the same, and false otherwise
Examples of Circuit Design and Construction
Compare-for-equality circuit
Addition circuit
Both circuits can be built using the circuit design algorithm
CE compares two unsigned binary integers for equality
Built by combining together 1-bit comparison circuits (1-CE)
Integers are equal if corresponding bits are equal (AND together 1-CD circuits for each pair of bits)
A Compare-for-Equality Circuit
1-CE circuit truth table
a b Output
0 0 1
0 1 0
1 0 0
1 1 1
A Compare-for-Equality Circuit (continued)
1-CE Boolean expression
First case: (NOT a) AND (NOT b)
Second case: a AND b
Combined:
((NOT a) AND (NOT b)) OR (a AND b)
A Compare-for-Equality Circuit (continued)
Figure 4.22One-Bit Compare-for-Equality Circuit
N-Bit Compare for Equality Circuit
AND together the 1-CE circuits, two at a time
An Addition CircuitAdds two unsigned binary integers, setting output bits and an overflow
Built from 1-bit adders (1-ADD)
Starting with rightmost bits, each pair produces
A value for that order
A carry bit for next place to the left
1-ADD truth table
Input
One bit from each input integer
One carry bit (always zero for rightmost bit)
Output
One bit for output place value
One carry bit
An Addition Circuit (continued)
Figure 4.24The 1-ADD Circuit and Truth Table
Building the full adder
Put rightmost bits into 1-ADD, with zero for the input carry
Send 1-ADD’s output value to output, and put its carry value as input to 1-ADD for next bits to left
Repeat process for all bits See pg 174, 175, 176
An Addition Circuit (continued)