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Discrete Mathematical Structures Homework Solution - CS173 Spring 2007, Assignments of Discrete Structures and Graph Theory

University of Illinois - Urbana-Champaign Discrete Structures and Graph Theory

Solutions to problem sets from a discrete mathematical structures course offered by the university of california, berkeley in spring 2007. The problems cover topics such as sets, functions, and equations.

Typology: Assignments

Pre 2010

Uploaded on 03/16/2009

koofers-user-1rf 🇺🇸

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CS173 Discrete Mathematical Structures

Spring 2007

Homework #4

Due Sun 02/11/07, 8AM.

Problem 1: (8 Points, 2 points each part)

Let Ai = {i, i+1, i+2,…} ⊆ P(N). Find

1.

! "

4

1 1= = !

!

"

#

$

%

&

i

j

A

2.

! "

n

i

j

A

1 1= = !

!

"

#

$

%

&

3.

( )

! "

n

j

i

A

1 1= = !

!

"

#

$

%

&

4.

( )

! !

n

j

i

A

1 1!="

"

#

$

%

&

'

Solution:

1.

! "

4

1 1= = !

!

"

#

$

%

&

i

j

A

=

! """! ""! " !

!

"

#

$

%

&

!

"

#

$

%

&

!

"

#

$

%

&

4321321211 AAAAAAAAAA

={1,2,3,…}

2. This is just the more general case of (1), which has the same answer: {1,2,3,…}. You can prove this by

proving first that

!

i

j

jAA

1

=

!

, and pointing out that

1

A

union with any subset of

1

A

is

1

A

.

3. Just like in the (1)-(2) case, let ’s start on a few numbers to get a feeling.

( ) ( ) ( )

! ""! "! " ,...}3,2,1{

321211

3

1 1

==

!

"

#

$

%

&

= =

AAAAAAA

j

i

Generalizing this result, we get: {1,2,3,…}.

4. Use the generalized De-Morgan law:

( ) ! "! !

n

j

i

n

j

i

iAA

1 11 1 !=!="

"

#

$

%

&

'

=

"

#

$

%

&

'

Let us denote

Bj=Ai=A1={}

i=1

j

!

Therefore,

Ai

( )

i=1

j

!

"

#$

%

&

j'1

n

!=Bj

j'1

n

!={}

Problem 2: (14 points: 10 points for writing a system of equations, 4 points for solving it)

An elite exhibition tournament featuring the Yankees, the Sox and the Cubs is staged in Japan.

35,000 tickets have been sold for the final game.

Among the fans, there are:

20,000 spectators who just want to eat a hotdog in the stadium and do not favor any team.

15,000 fans root for one of the Chicago teams (note: some may root for both of them!).

Discover Assignments of Discrete Structures and Graph Theory University of Illinois - Urbana-Champaign

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Download Discrete Mathematical Structures Homework Solution - CS173 Spring 2007 and more Assignments Discrete Structures and Graph Theory in PDF only on Docsity!

CS173 Discrete Mathematical Structures

Spring 200 7

Homework

Due Sun 02/11/0 7 , 8AM.

Problem 1: (8 Points, 2 points each part)

Let Ai = { i , i +1, i +2,…} ⊆ P(N). Find

4

= 1 = 1

i

j

A

n

i

j

A

= 1 = 1

n

j

i

A

= 1 = 1

n

j

i

A

! 1 = 1

Solution:

4

= 1 = 1

i

j

A =

1 1 2 1 2 3 1 2 3 4

A A A A A A A A A A ={1,2,3,…}

This is just the more general case of (1), which has the same answer: {1,2,3,…}. You can prove this by

proving first that

i

j

A j A

1

=

(^)! , and pointing out that A 1 union with any subset of A 1 is A 1.

Just like in the (1)-(2) case, let’s start on a few numbers to get a feeling.

1 1 2 1 2 3

3

1 1

= =

A A A A A A A

j

i

Generalizing this result, we get: {1,2,3,…}.

4. Use the generalized De-Morgan law: ( )

n

j

i

n

j

i

A A

! 1 = 1! 1 = 1

Let us denote Bj =^ Ai =^ A 1 =^ {}

i = 1

j

Therefore,

A

i

i = 1

j

j ' 1

n

! =^ Bj

j ' 1

n

! =^ {}

Problem 2: (14 points: 10 points for writing a system of equations, 4 points for solving it)

An elite exhibition tournament featuring the Yankees, the Sox and the Cubs is staged in Japan.

35,000 tickets have been sold for the final game.

Among the fans, there are:

20,000 spectators who just want to eat a hotdog in the stadium and do not favor any team.

15,000 fans root for one of the Chicago teams (note: some may root for both of them!).

13,000 fans root for the Yankees or for the Sox (note: some may root for both of them!).

7000 root for the Sox only.

1000 cheerful fans root for all the three teams

How many fans root for both Yankees and the Cubs.

Solution:

Solve the following system of equations:

35,000-20,000=|Y| + |S| + |C| - |Y∩C| - |Y∩S| - |S∩C| + |Y∩S∩C|

15,000=|S| + |C| - |S∩C|

13,000=|S| + |Y| - |Y∩S|

7,000=|S|

1000=|Y∩S∩C|

We get:

15,000=(|Y| + |S| - |Y∩S| )+ (|C| + |S| - |S| - |S∩C| ) - |Y∩C| + |Y∩S∩C|

15,000=(|Y| + |S| - |Y∩S| )+ (|C| +|S| - |S∩C| ) - |Y∩C| + |Y∩S∩C|-|S|

15,000=15,000+13,000- |Y∩C| + 1,000-7,

|Y∩C| =7,

Problem 3: (8 points, 1 point for each part)

Determine whether the following are functions:

f : R! R where

1 /( 1 ) otherwise

1 / if 0

( )

x

x x

f x

f : R! R where f ( x )= x
f : N! R where f ( x )= x
f : R! R where

2 f ( x )=± x

f : Z! Z where

2 f ( x )=+ x

f : N! Z where f ( x )=+ x
f : Z! R where

2 f ( x )=+ x

f : N! R where

6 8

2

x x

f x

Solutions:

This is a function.
Not a function- not defined at x<
This is a function.
Not a function, since a function cannot have two values at the same point
This is a function.
Not a function, since a root of a positive integer is not necessarily integer. (f(5) is not defined)
This is a function
This is a function, because the denominator is 0 only at points – 2,-4, which are not in the

Discrete Mathematical Structures Homework Solution - CS173 Spring 2007, Assignments of Discrete Structures and Graph Theory

Related documents

Partial preview of the text

Download Discrete Mathematical Structures Homework Solution - CS173 Spring 2007 and more Assignments Discrete Structures and Graph Theory in PDF only on Docsity!

CS173 Discrete Mathematical Structures

Spring 200 7

Homework

Due Sun 02/11/0 7 , 8AM.

A

A

A

A

A =

A A A A A A A A A A ={1,2,3,…}

A A A A A A A

4. Use the generalized De-Morgan law: ( )

A A

A

! =^ Bj

! =^ {}

35,000-20,000=|Y| + |S| + |C| - |Y∩C| - |Y∩S| - |S∩C| + |Y∩S∩C|

15,000=|S| + |C| - |S∩C|

13,000=|S| + |Y| - |Y∩S|

7,000=|S|

1000=|Y∩S∩C|

15,000=(|Y| + |S| - |Y∩S| )+ (|C| + |S| - |S| - |S∩C| ) - |Y∩C| + |Y∩S∩C|

15,000=(|Y| + |S| - |Y∩S| )+ (|C| +|S| - |S∩C| ) - |Y∩C| + |Y∩S∩C|-|S|

15,000=15,000+13,000- |Y∩C| + 1,000-7,

|Y∩C| =7,