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This was adjacency matrix after k=x and k=1 steps. Notice the shorter (green) path from x to 4.

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This was adjacency matrix after k=x and k=1 steps. Notice the shorter (green) path from x to 4. - PowerPoint PPT Presentation


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1. 3. 2. 5. 1. 2. 1. x. 2. This was adjacency matrix after k=x and k=1 steps. Notice the shorter (green) path from x to 4. Now use k=3 and see how we pick it up. We do get to look at paths of length > 2!. 4. 4. 3. 3. X 1 2 3 4. X 1 2 3 4.

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Presentation Transcript
slide1
1

3

2

5

1

2

1

x

2

This was adjacency matrix after k=x and k=1 steps. Notice the shorter (green) path from x to 4.

Now use k=3 and see how we pick it up. We do get to look at paths of length > 2!

4

4

3

3

X 1 2 3 4

X

1

2

3

4

A[i,3]+A[3,j] < A[i,j] ?

7 2 logic circuits
7.2 Logic circuits
  • Boolean Expressions
  • Logic Networks
  • Truth Functions
  • Can every truth table be written as a Boolean expression? As a computer circuit?
switches
Switches

Control

(off=false)

+

-

Control

(on=true)

+

-

a or b
A Or B

A

+

-

B

and or and not
And, Or and Not

And ( . )

Or ( + )

Not (‘)

boolean expressions
Boolean Expressions
  • A Boolean Expression in n variables x1,x2, .. Xn, is any string of finite symbols formed by the following rules
  • 1. x1,x2, .. Xn are Boolean expressions
  • 2. If P and Q are Boolean Expressions, then so are (P+Q), PQ , and P’.
boolean expressions1
Boolean Expressions

(x + y) x + y’(xy+z)

is a Boolean expression.

truth functions
Truth Functions
  • A truth function is a truth table for Boolean expressions
find a network for xy z
Find a network for xy’+z

X

Y

Z

Exercise: Find a network for (xy’+z)’

slide13
We can do the following

Truth function Boolean expression

Boolean expression logic network

Can we find a Boolean expression for any truth function? Yes!

canonical form
Canonical Form
  • For the truth function shown, find a Boolean expression:

True for:

xyz’

xy’z

x’yz

x’yz’

x’y’z

slide15
So the truth function is given by:

xyz’+xy’z+x’yz+x’yz’+x’y’z

This is the Canonical form for the truth function (a standard form into which we can place any two Boolean expressions for comparison.)

slide16
Simplify, using Boolean algebra:

xyz’+xy’z+x’yz+x’yz’+x’y’z

= xyz’+xy’z+x’yz+x’yz’+x’y’z+x’yz’

=y’z(x+x’)+x’y(z+z’)+yz’(x+x’)

=y’z+x’y+yz’

Notice the groupings of two the same/ one different.

half adder
Half Adder
  • Design a circuit to add binary numbers
half adder as two functions
Half Adder-as two functions

xy’+x’y=(x+y)(xy)’

xy

the half adder
The half-adder

(x+y)(xy)’

x

sum

y

x

y

carry

xy

to add two binary numbers
To add two binary numbers

Full adder

Cprevious

s

Half adder

x

c

s

cnext

Half Adder

y

c

general add xyz abc
General add: xyz+abc

x

Digit 1

Half adder

a

Digit 2

Full adder

y

b

Digit 3

Full adder

z

c

Digit 4

ad