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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

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

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Lecture notes

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


A and b

A and B

A

B

+

-


Not a

Not A

A

-

+


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)’


Find a boolean expression

Find a Boolean expression

x

y

?

z


Lecture notes

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


Lecture notes

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.)


Lecture notes

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


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