Cs 173 discrete mathematical structures
This presentation is the property of its rightful owner.
Sponsored Links
1 / 20

CS 173: Discrete Mathematical Structures PowerPoint PPT Presentation


  • 81 Views
  • Uploaded on
  • Presentation posted in: General

CS 173: Discrete Mathematical Structures. Cinda Heeren [email protected] Siebel Center, rm 2213 Office Hours: W 9:30-11:30a. CS 173 Announcements. Homework 7 available. Due 03/12, 8a. Exam 2, Apr 4, 7-9p, Loomis 141. Email Cinda with conflicts.

Download Presentation

CS 173: Discrete Mathematical Structures

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Cs 173 discrete mathematical structures

CS 173:Discrete Mathematical Structures

Cinda Heeren

[email protected]

Siebel Center, rm 2213

Office Hours: W 9:30-11:30a


Cs 173 announcements

CS 173 Announcements

  • Homework 7 available. Due 03/12, 8a.

  • Exam 2, Apr 4, 7-9p, Loomis 141. Email Cinda with conflicts.

  • Today’s lecture covers material from Rosen, sections 4.1-4.3.

Cs173 - Spring 2004


Cs173 decision tree

CS173Decision Tree

Suppose you have 4 shirts, 3 pairs of pants, and 2 pairs of shoes. How many different outfits do you have?

Cs173 - Spring 2004


Cs173 decision tree1

20

A DT is a good model for a sequence of events. It assists in counting, and can help you see special structure in the problem.

CS173 Decision Tree

How many different best of 5 game series are possible between the Cardinals and the Astros?

Cs173 - Spring 2004


Cs173 pigeonhole principle

CS173Pigeonhole Principle

If n pigeons fly into k pigeonholes and k < n, then some pigeonhole contains at least two pigeons.

Cs173 - Spring 2004


Cs173 pigeonhole principle1

We can use this simple little fact to prove amazingly complex things.

CS173Pigeonhole Principle

If n pigeons fly into k pigeonholes and k < n, then some pigeonhole contains at least two pigeons.

Cs173 - Spring 2004


Cs173 pigeonhole principle2

0, 1, 2, 3, or 4

Some pair has the same remainder, by PHP.

CS173Pigeonhole Principle

Let S contain any 6 positive integers. Then, there is a pair of numbers in S whose difference is divisible by 5.

Let S = {a1,a2,a3,a4,a5,a6}. Each of these has a remainder when divided by 5. What can these remainders be?

6 numbers, 5 possible remainders…what do we know?

Consider that pair, ai and aj, and their remainder r.

ai = 5m + r, and aj = 5n + r.

Their difference: ai - aj = (5m + r) - (5n + r) = 5m - 5n = 5(m-n), which is divisible by 5.

Cs173 - Spring 2004


Cs173 pigeonhole principle3

Let’s say she knows 3 others.

Consider one person.

CS173Pigeonhole Principle

Six people go to a party. Either there is a group of 3 who all know each other, or there is a group of 3 who are all strangers.

If any of those 3 know each other, we have a blue , which means 3 people know each other. So they all must be strangers.

She either knows or doesn’t know each other person.

But then we’ve proven our conjecture for this case.

But there are 5 other people! So, she knows, or doesn’t know, at least 3 others.

The case where she doesn’t know 3 others is similar.

Cs173 - Spring 2004


Cs 199 ice cream cones

CS 199 Ice Cream Cones

Do these two cones provide the same ice cream experience?

Cs173 - Spring 2004


Cs173 permutations

P(n,r) = n! / (n-r)!

  • 60

  • 125

  • 12!/7!

  • 512

  • No clue

CS173 Permutations

In a running race of 12 sprinters, each of the top 5 finishers receives a different medal. How many ways are there to award the 5 medals?

11

10

9

8

12

A permutation is an ordered arrangement of objects.

The number of permutations of r distinct objects chosen from n distinct objects is denoted P(n,r).

Cs173 - Spring 2004


Cs173 permutations1

P(17,10) = 17x16x15x14x13x12x11

CS173 Permutations

Suppose you have time to listen to 10 songs on your daily jog around campus. There are 6 Cake tunes, 8 Moby tunes, and 3 Eagles tunes to choose from.

How many different jog playlists can you make?

Cs173 - Spring 2004


Cs173 permutations2

P(6,4) x P(8,4) x P(3,2)

CS173 Permutations

Suppose you have time to listen to 10 songs on your daily jog around campus. There are 6 Cake tunes, 8 Moby tunes, and 3 Eagles tunes to choose from.

Now suppose you want to listen to 4 Cake, 4 Moby, and 2 Eagles tunes, in that band order. How many playlists can you make?

Cs173 - Spring 2004


Cs173 permutations3

P(6,4) x P(8,4) x P(3,2) x 3!

CS173 Permutations

Suppose you have time to listen to 10 songs on your daily jog around campus. There are 6 Cake tunes, 8 Moby tunes, and 3 Eagles tunes to choose from.

Finally, suppose you still want 4 Cake, 4 Moby, and 2 Eagles tunes, and the order of the groups doesn’t matter, but you get dizzy and fall down if all the songs by any one group aren’t played together.

How many playlists are there now?

Cs173 - Spring 2004


Cs173 permutations4

M2

M4

M5

M1

M3

5! X P(6,3)

CS173 Permutations

In how many ways can 5 distinct Martians and 3 distinct Jovians stand in line, if no two Jovians stand together?

Cs173 - Spring 2004


Cs173 combinations

CS173 Combinations

A combination is an unordered selection of elements from some set.

The number of combinations of r distinct objects chosen from n distinct objects is denoted by C(n,r) or nCr or , and is read “n choose r.”

C(n,r) = P(n,r)/r! = n!/((n-r)!r!)

Cs173 - Spring 2004


Cs173 combinations1

C(12,5)

P(12,5)

= C(12,5) x 5!

CS173Combinations

A basketball squad consists of 12 players, 5 of which make up a team. How many different teams of players can you make from the 12?

What’s the diff?

In a running race of 12 sprinters, each of the top 5 finishers receives a different medal. How many ways are there to award the 5 medals?

Cs173 - Spring 2004


Cs173 combinations2

CS173 Combinations

A committee of 8 students is to be selected from a class consisting of 19 frosh, and 34 soph.

In how many ways can 3 frosh and 5 soph be selected?

Cs173 - Spring 2004


Cs173 combinations3

CS173 Combinations

A committee of 8 students is to be selected from a class consisting of 19 frosh, and 34 soph.

In how many ways can a committee with exactly 1 frosh be selected?

Cs173 - Spring 2004


Cs173 combinations4

CS173 Combinations

A committee of 8 students is to be selected from a class consisting of 19 frosh, and 34 soph.

In how many ways can a committee with at most 1 frosh be selected?

Cs173 - Spring 2004


Cs173 combinations5

CS173 Combinations

A committee of 8 students is to be selected from a class consisting of 19 frosh, and 34 soph.

In how many ways can a committee with at least 1 frosh be selected?

Cs173 - Spring 2004


  • Login