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Binary logic. Binary logic is a mathematical system that lets us reason about logic statements. IF The garage door is open AND The engine is running THEN The car can be backed out of the garage. The car can be backed out only when both conditions are true.

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binary logic
Binary logic

Binarylogic is a mathematical system that lets us reason about logic statements

IFThe garage door is open

AND The engine is runningTHEN The car can be backed out of the garage

The car can be backed

out only when both

conditions are true

The light will become yellow only if it’s been green for > 45 seconds or nobody is on the road

IFTheN-S light is green

AND The E-W light is red

AND (The N-S light has been green for more than 45 sec.OR There are no cars on the N-S road)THEN The N-S lights can be changed from green to yellow

combinational logic

Door Open? Engine Running? OK to Back Out

False False False

False True False

True False False

True True True

Combinational Logic

Each input can beeither True or False

IFThe garage door is open

AND The engine is runningTHEN The car can be backed out of the garage

What is the output for each combination of inputs?

There are 2N combinations to be considered for N binary inputs.

truth tables

X Y X and Y

X Y X or Y

X not X

F T

T F

F F F

F T T

T F T

T T T

F F F

F T F

T F F

T T T

Input

Output

Truth tables
  • Truth tables enumerate all possible input combinations
    • For each input, tabulate the output
    • There may be more than one independent output
  • A truth table that enumerates all input combinationscompletely defines any logic function

For n inputs: 2n rows

the binary connection

X Y X and Y

X Y X or Y

X not X

0 1

1 0

0 0 0

0 1 1

1 0 1

1 1 1

0 0 0

0 1 0

1 0 0

1 1 1

Input

Output

The Binary Connection
  • Truth or Falsehood is a Binary operation
    • Everything is either True or False, no in-betweens
  • Represent True using ‘1’
  • Represent False using ‘0’

Note: Number combinations in binary numeric order:

00, 01, 10, 11

2.2

example
Example

a b c d FG

0 0 0 0 1 00 0 0 1 0 10 0 1 0 0 1

0 0 1 1 1 1

0 1 0 0 0 0

0 1 0 1 1 0

0 1 1 0 1 0

0 1 1 1 0 0

1 0 0 0 0 0

1 0 0 1 1 0

1 0 1 0 1 0

1 0 1 1 0 0

1 1 0 0 1 0

1 1 0 1 0 0

1 1 1 0 0 0

1 1 1 1 1 0

  • Function F(a,b,c,d) should be 1 whenever there are an even number of inputs that are 1
  • Function G(a,b,c,d) should be 1 whenever c is 1 or d is 1, but not when a or b is 1
illegal inputs

M

F

T

S

Good

0

0

0

0

0

0

0

1

0

0

1

0

0

0

1

1

0

1

0

0

0

1

0

1

0

1

1

0

0

1

1

1

1

0

0

0

1

0

0

1

1

0

1

0

1

0

1

1

1

1

0

0

1

1

0

1

1

1

1

0

1

1

1

1

Illegal Inputs
  • The women’s basketball team is looking for good players (women 5’9” or taller)
    • The data available is:
      • M: True if male
      • F: True if female
      • T: True if 5’9” or taller
      • S: True if < 5’9”

X

X

X

X

X

0

1

X

X

  • Many combinations are impossible
    • Can’t be Male and Female
    • Can’t be Tall and Short

0

0

X

X

  • Impossible input combinations are marked with an ‘X’
    • Called a don’t care

X

X

X

logic primitives

x’

x

not(x)

x

xy

x and y

y

x

x or y

x+y

y

precedence rules

Logic Primitives

NOT before AND before OR

timing diagram

T1

A

Z

B

T2

C

1

A

0

0

0

0

1

1

1

1

0

1

B

0

0

1

1

0

0

1

1

0

1

C

0

1

0

1

0

1

0

1

0

1

T1

1

1

1

1

0

0

0

0

0

1

T2

0

1

1

1

0

1

1

1

0

1

Z

0

1

1

1

0

0

0

0

0

Timing diagram
  • A timing diagram may be used to express the behavior of a logic system

A B C T1 T2 Z

0 0 0 1 0 0

0 0 1 1 1 1

0 1 0 1 1 1

0 1 1 1 1 1

1 0 0 0 0 0

1 0 1 0 1 0

1 1 0 0 1 0

1 1 1 0 1 0

Inputs

functions of two variables

F8

F9

F10

F1

1

F12

F13

F14

F15

X

Y

F0

F1

F2

F3

F4

F5

F6

F7

1

1

1

1

1

1

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

1

1

1

0

1

0

0

0

0

1

1

1

1

0

0

1

1

0

0

1

1

1

0

0

0

1

1

0

0

1

1

0

1

0

1

0

1

0

1

1

1

0

1

0

1

0

1

0

1

X

Y

X

+

Y

Functions of two variables

0

1

X

Y

X

Y

There are sixteen functions of two variables…We’ve only seen eight of them so far

nands and nors

X nand Y = not (X and Y) =

X Y Z

0 0 1

0 1 1

1 0 1

1 1 0

X nor Y = not (X or Y) =

X Y Z

0 0 1

0 1 0

1 0 0

1 1 0

NANDs and NORs
xors and xnors

Exclusive OR - XOR

XOR - True if both inputs

are different

X Y Z

0 0 0

0 1 1

1 0 1

1 1 0

Equivalence gate - XNOR

X Y Z

0 0 1

0 1 0

1 0 0

1 1 1

XNOR - True if both inputs

are the same

XORs and XNORs
what s left

F8

F9

F10

F1

1

F12

F13

F14

F15

X

Y

F0

F1

F2

F3

F4

F5

F6

F7

1

1

1

1

1

1

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

1

1

1

0

1

0

0

0

0

1

1

1

1

0

0

1

1

0

0

1

1

1

0

0

0

1

1

0

0

1

1

0

1

0

1

0

1

0

1

1

1

0

1

0

1

0

1

0

1

X

Y

X

+

Y

What’s left?

0

1

X

Y

X

Y

Remaining functions are implication functions, which aren’t

commonly used