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CS 173: Discrete Mathematical Structures. Cinda Heeren [email protected] Rm 2213 Siebel Center Office Hours: W 12:30-2:30p. What homework? I’ve read it. 25% done 50% done 75% done. CS 173 Announcements. Homework 2 returned this week. Homework 3 available. Due 09/16.

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Cs 173 discrete mathematical structures

CS 173:Discrete Mathematical Structures

Cinda Heeren

[email protected]

Rm 2213 Siebel Center

Office Hours: W 12:30-2:30p


Cs 173 announcements

CS 173 Announcements

  • Homework 2 returned this week.

  • Homework 3 available. Due 09/16.

  • LaTex workshop Th, 9/15, at 6:30p in SC 2405.

  • The following reflects my status on hwk #3…

Cs173 - Spring 2004


Cs 173 set theory cardinality

|S| = 3.

|S| = 1.

|S| = 0.

|S| = 3.

CS 173 Set Theory - Cardinality

If S is finite, then the cardinality of S, |S|, is the number of distinct elements in S.

If S = {1,2,3},

If S = {3,3,3,3,3},

If S = ,

If S = { , {}, {,{}} },

If S = {0,1,2,3,…}, |S| is infinite. (more on this later)

Cs173 - Spring 2004


Cs 173 set theory power sets

aka P(S)

We say, “P(S) is the set of all subsets of S.”

2S = {, {a}}.

2S = {, {a}, {b}, {a,b}}.

2S = {}.

2S = {, {}, {{}}, {,{}}}.

CS 173 Set Theory - Power sets

If S is a set, then the power set of S is

2S = { x : x  S }.

If S = {a},

If S = {a,b},

If S = ,

If S = {,{}},

Fact: if S is finite, |2S| = 2|S|. (if |S| = n, |2S| = 2n)

Cs173 - Spring 2004


Cs 173 set theory cartesian product

A,B finite  |AxB| = ?

We’ll use these special sets soon!

AxB

|A|+|B|

|A+B|

|A||B|

CS 173 Set Theory - Cartesian Product

The Cartesian Product of two sets A and B is:

A x B = { <a,b> : a  A  b  B}

If A = {Charlie, Lucy, Linus}, and B = {Brown, VanPelt}, then

A x B = {<Charlie, Brown>, <Lucy, Brown>, <Linus, Brown>, <Charlie, VanPelt>, <Lucy, VanPelt>, <Linus, VanPelt>}

A1 x A2 x … x An = {<a1, a2,…, an>: a1  A1, a2  A2, …, an  An}

Cs173 - Spring 2004


Cs 173 set theory operators

B

A

CS 173 Set Theory - Operators

The union of two sets A and B is:

A  B = { x : x  A v x  B}

If A = {Charlie, Lucy, Linus}, and B = {Lucy, Desi}, then

A  B = {Charlie, Lucy, Linus, Desi}

Cs173 - Spring 2004


Cs 173 set theory operators1

B

A

CS 173 Set Theory - Operators

The intersection of two sets A and B is:

A  B = { x : x  A  x  B}

If A = {Charlie, Lucy, Linus}, and B = {Lucy, Desi}, then

A  B = {Lucy}

Cs173 - Spring 2004


Cs 173 set theory operators2
CS 173 Set Theory - Operators

The intersection of two sets A and B is:

A  B = { x : x  A  x  B}

If A = {x : x is a US president}, and B = {x : x is deceased}, then

A  B = {x : x is a deceased US president}

B

A

Cs173 - Spring 2004


Cs 173 set theory operators3

B

A

Sets whose intersection is empty are called disjoint sets

CS 173 Set Theory - Operators

The intersection of two sets A and B is:

A  B = { x : x  A  x  B}

If A = {x : x is a US president}, and B = {x : x is in this room}, then

A  B = {x : x is a US president in this room} = 

Cs173 - Spring 2004


Cs 173 set theory operators4

A

  • = U and

    U = 

CS 173 Set Theory - Operators

The complement of a set A is:

A = { x : x  A}

If A = {x : x is bored}, then

A = {x : x is not bored}

= 

U

Cs173 - Spring 2004


Cs 173 set theory operators5

U

A

B

CS 173 Set Theory - Operators

The set difference, A - B, is:

A - B = { x : x  A  x  B }

A - B = A  B

Cs173 - Spring 2004


Cs 173 set theory operators6

U

A

B

CS 173 Set Theory - Operators

The symmetric difference, A  B, is:

A  B = { x : (x  A  x  B) v (x  B  x  A)}

= (A - B) U (B - A)

Cs173 - Spring 2004


Cs 173 set theory operators7
CS 173 Set Theory - Operators

A  B = { x : (x  A  x  B) v (x  B  x  A)}

= (A - B) U (B - A)

Proof:

{ x : (x  A  x  B) v (x  B  x  A)}

= { x : (x  A - B) v (x  B - A)}

= { x : x  ((A - B) U (B - A))}

= (A - B) U (B - A)

Cs173 - Spring 2004


Cs 173 set theory famous identities

Don’t memorize them, understand them!

They’re in Rosen, p89.

CS 173 Set Theory - Famous Identities

  • Two pages of (almost) obvious.

  • One page of HS algebra.

  • One page of new.

Cs173 - Spring 2004


Cs 173 set theory famous identities1

A  U = A

A U U = U

A U A = A

A U  = A

A  = A

A A = A

(Lazy)

CS 173 Set Theory - Famous Identities

  • Identity

  • Domination

  • Idempotent

Cs173 - Spring 2004


Cs 173 set theory famous identities2

A U A = U

A = A

A A=

CS 173 Set Theory - Famous Identities

  • Excluded Middle

  • Uniqueness

  • Double complement

Cs173 - Spring 2004


Cs 173 set theory famous identities3

B U A

B  A

A U (B U C)

A U (B C) =

A  (B C)

A  (B U C) =

CS 173 Set Theory - Famous Identities

  • Commutativity

  • Associativity

  • Distributivity

A U B =

A  B =

(A U B)U C =

(A  B) C =

(A U B)  (A U C)

(A  B) U (A  C)

Cs173 - Spring 2004


Cs 173 set theory famous identities4

(A UB)= A  B

(A  B)= A U B

Hand waving is good for intuition, but we aim for a more formal proof.

CS 173 Set Theory - Famous Identities

  • DeMorgan’s I

  • DeMorgan’s II

p

q

Cs173 - Spring 2004


Cs 173 set theory 4 ways to prove identities

New & important

Like truth tables

Like 

Not hard, a little tedious

CS 173 Set Theory - 4 Ways to prove identities

  • Show that A  B and that A  B.

  • Use a membership table.

  • Use previously proven identities.

  • Use logical equivalences to prove equivalent set definitions.

Cs173 - Spring 2004


Cs 173 set theory 4 ways to prove identities1

(A UB)= A  B

CS 173 Set Theory - 4 Ways to prove identities

Prove that

  • () (x  A U B)  (x  A U B)  (x  A and x  B)  (x  A  B)

    2. () (x  A  B)  (x  A and x  B)  (x  A U B)  (x  A U B)

Cs173 - Spring 2004


Cs 173 set theory 4 ways to prove identities2

(A UB)= A  B

Haven’t we seen this before?

CS 173 Set Theory - 4 Ways to prove identities

Prove that using a membership table.

0 : x is not in the specified set

1 : otherwise

Cs173 - Spring 2004


Cs 173 set theory 4 ways to prove identities3

(A UB)= A  B

(A UB)= A U B

= A  B

= A  B

CS 173 Set Theory - 4 Ways to prove identities

Prove that using identities.

Cs173 - Spring 2004


Cs 173 set theory 4 ways to prove identities4

(A UB)= A  B

(A UB)= {x : (x  A v x  B)}

= A  B

= {x : (x  A)  (x  B)}

CS 173 Set Theory - 4 Ways to prove identities

Prove that using logically equivalent set definitions.

= {x : (x  A)  (x  B)}

Cs173 - Spring 2004


Cs 173 set theory a proof for us to do together
CS 173 Set Theory - A proof for us to do together.

X  (Y - Z) = (X  Y) - (X  Z). True or False?

Prove your response.

(X  Y) - (X  Z) = (X  Y)  (X  Z)’

= (X  Y)  (X’ U Z’)

= (X  Y  X’) U (X  Y  Z’)

=  U (X  Y  Z’)

= (X  Y  Z’)

= X  (Y - Z)

Cs173 - Spring 2004


Cs 173 set theory a proof for us to do together1

A U B =

A = B

A  B = 

A-B = B-A = 

Trying to pv p --> q

Assume p and not q, and find a contradiction.

Our contradiction was that sets weren’t equal.

CS 173 Set Theory - A proof for us to do together.

Pv that if (A - B) U (B - A) = (A U B) then ______

A  B = 

Suppose to the contrary, that A  B  , and that x  A  B.

Then x cannot be in A-B and x cannot be in B-A.

DeMorgan’s!!

Then x is not in (A - B) U (B - A).

Do you see the contradiction yet?

But x is in A U B since (A  B)  (A U B).

Thus, A  B = .

Cs173 - Spring 2004


Cs 173 set theory generalized union
CS 173 Set Theory - Generalized Union

Ex. Let U = N, and define:

A1 = {2,3,4,…}

A2 = {2,4,6,…}

A3 = {3,6,9,…}

Cs173 - Spring 2004


Cs 173 set theory generalized union1

Primes

Composites

N

I have no clue.

primes

CS 173 Set Theory - Generalized Union

Ex. Let U = N, and define:

Then

Cs173 - Spring 2004


Cs 173 set theory generalized intersection
CS 173 Set Theory - Generalized Intersection

Ex. Let U = N, and define:

A1 = {1,2,3,4,…}

A2 = {2,4,6,…}

A3 = {3,6,9,…}

Cs173 - Spring 2004


Cs 173 set theory generalized intersection1

Multiples of LCM(1,…,n)

CS 173 Set Theory - Generalized Intersection

Ex. Let U = N, and define:

Then

Cs173 - Spring 2004


Cs 173 set theory inclusion exclusion

B

A

Wrong.

CS 173 Set Theory - Inclusion/Exclusion

Example:

How many people are wearing a watch?

How many people are wearing sneakers?

How many people are wearing a watch OR sneakers?

What’s wrong?

|A  B| = |A| + |B| - |A  B|

Cs173 - Spring 2004


Cs 173 set theory inclusion exclusion1

125

173

217 - (157 + 145 - 98) = 13

CS 173 Set Theory - Inclusion/Exclusion

Example:

There are 217 cs majors.

157 are taking cs125.

145 are taking cs173.

98 are taking both.

How many are taking neither?

Cs173 - Spring 2004


Cs 173 set theory generalized inclusion exclusion

Now let’s do it for 4 sets!

kidding.

CS 173 Set Theory - Generalized Inclusion/Exclusion

Suppose we have:

B

A

C

And I want to know |A U B U C|

|A U B U C| = |A| + |B| + |C|

- |A  B| - |A  C| - |B  C|

+ |A  B  C|

Cs173 - Spring 2004


Cs 173 set theory inclusion exclusion2
CS 173 Set Theory - Inclusion/Exclusion

Example:

How many people are wearing a watch?

How many people are wearing sneakers?

How many people are wearing a watch AND sneakers?

How many people are wearing a watch OR sneakers?

Cs173 - Spring 2004


Cs 173 set theory generalized inclusion exclusion1
CS 173 Set Theory - Generalized Inclusion/Exclusion

For sets A1, A2,…An we have:

Cs173 - Spring 2004


Cs 173 set theory sets as bit strings

(101011)

CS 173 Set Theory - Sets as bit strings

Let U = {x1, x2,…, xn}, and let A  U.

Then the characteristic vector of A is the n-vector whose elements, xi, are 1 if xi A, and 0 otherwise.

Ex. If U = {x1, x2, x3, x4, x5, x6}, and A = {x1, x3, x5, x6}, then the characteristic vector of A is

Cs173 - Spring 2004


Cs 173 set theory sets as bit strings1

Bit-wise OR

Bit-wise AND

CS 173 Set Theory - Sets as bit strings

Ex. If U = {x1, x2, x3, x4, x5, x6}, A = {x1, x3, x5, x6}, and B = {x2, x3, x6},

Then we have a quick way of finding the characteristic vectors of A  B and A  B.

Cs173 - Spring 2004


Cs 173 functions

f(x) = -(1/2)x - 25

domain

co-domain

CS 173 Functions

Suppose we have:

And I ask you to describe the yellow function.

What’s a function?

Notation: f: RR, f(x) = -(1/2)x - 25

Cs173 - Spring 2004


Cs 173 functions1
CS 173 Functions

Definition: a function f : A  B is a subset of AxB where  a  A, ! b  B and <a,b>  f.

Cs173 - Spring 2004


Cs 173 functions2

B

A

A point!

A collection of points!

CS 173 Functions

Definition: a function f : A  B is a subset of AxB where  a  A, ! b  B and <a,b>  f.

B

A

Cs173 - Spring 2004


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