# CPSC 171 Introduction to Computer Science - PowerPoint PPT Presentation

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CPSC 171 Introduction to Computer Science. Boolean Logic, Gates, & Circuits. Announcements. Read Chapter 4 Exam, Oct 2 nd 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

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CPSC 171 Introduction to Computer Science

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### CPSC 171 Introduction to Computer Science

Boolean Logic, Gates, & Circuits

### Announcements

• 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

### Boolean Logic (continued)

• Example:

(a AND b) OR ((NOT b) and (NOT a))

### 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 Circuit

1

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

• 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

• Adds two unsigned binary integers, setting output bits and an overflow

• Starting with rightmost bits, each pair produces

• A value for that order

• A carry bit for next place to the left

• 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