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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 false
- A Boolean expression, (A AND B), evaluates to either true or false
- Boolean operators include:
- AND (& • )
- OR ( + )
- NOT (a bar ' ¬ ~)

Boolean Operators

- a AND b

true only when A and B are both true

- a OR b

true when A is true, B is true, or both are true

- NOT a

true when A is false

Truth Tables

Truth tables can be used to capture when an expression is true, given its inputs

You make truth tables for AND and NOT

Example Boolean Expressions

(a AND b) OR (NOT a AND c)

a·b + ~a·c

ab+āc

Truth tables can be made for complex expressions as well

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 4.15

The Three Basic Gates and Their Symbols

Circuits

- A collection of logic gates that transforms a set of binary inputs into a set of binary outputs
- Wire gates together keeping constraints for the number of inputs to any gate

a

b

c

d

Example Circuit1

1

0

- If a, b, c, and d are all true the output can be determined by tracing through the circuit

output

1

0

1

1

1

Designing Circuits

A circuit construction algorithm

- Truth Table Construction

Determine outputs for every possible input

- 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

- Sub-expression combination (using OR gates)

Take each subexpression and combine them, 2 at a time, using OR gates

- Circuit Diagram Production

Construct 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

A Compare-for-Equality Circuit

- 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 (continued)

- 1-CE circuit truth table

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)

Figure 4.22

One-Bit Compare-for-Equality Circuit

N-Bit Compare for Equality Circuit

- AND together the 1-CE circuits, two at a time

An Addition Circuit

- Adds 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

An Addition Circuit (continued)

- 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

Figure 4.24

The 1-ADD Circuit and Truth Table

An Addition Circuit (continued)

- 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

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