### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# Prb&Rand Proc EECS 501

UM

GPA 3.8

### View Full Document

## 13

## 0

## Popular in Course

## Popular in Engineering Computer Science

This 1 page Class Notes was uploaded by Ophelia Ritchie on Thursday October 29, 2015. The Class Notes belongs to EECS 501 at University of Michigan taught by Staff in Fall. Since its upload, it has received 13 views. For similar materials see /class/231536/eecs-501-university-of-michigan in Engineering Computer Science at University of Michigan.

## Reviews for Prb&Rand Proc

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 10/29/15

EECS 501 COUNTABLE VS UNCOUNTABLE SETS Fall 2001 DEF A set is nite if it has a nite number of elements DEF Two sets A7 B are in one to one correspondence 771 177 if there exists a 1 1 mapping between elements of A and elements of B NOTE Two nite sets are 1 1 IFF they have same number of elements EX a7 b7 0 and 1017 102 126 are 1 1 26 elements each NOTE An in nite set can be 1 1 with a proper subset of itself A 1727 374 and B 274767 8 are 1 1 Mapping b 2a Z 27 1707172 andY Z1L 17273 are 1 1 1 1 Mapping z y2 if y is even 2 1 y2 if y is odd DEF A set is countably in nite IFF it is 1 1 with integers ie You can 77count77 the elements of the set this may take foreverl EX even integers and odd integers are countably in nite DEF A set is countable IFF it is either nite or countably in nite NOTE A set is countable IFF it is 1 1 with another countable set THM The set of lattice points 22 i7j i7j E integers is countable Proof A 1 1 mapping between i7j i 07 17 2 7j 07 17 2 and nn17273isnij1ij22 j Can easily extend to negative values as shown above THM The set of Rationals Q i7j E integersj gt 0 is countable Proof 1 1 Mapping Q 3 q ij lt gt i7j E 22 is known to be countable In fact7 Q is 1 1 with a subset of ZQ Q is at most countable BUT Countable Z C Q7 so Q is at least countable gt Q is exactly countable THM A countable union of countable sets is countable Proof The 77countable union77 is 1 1 with 2 reindex it with i 17 2 The 77countable sets77 can similarly each be reindexed with j 17 2 UfiiAz U291ai17ai2 U521 U511 am lt M 6 32 Again7 possible duplications gtthis is at most countable This theorem is particularly useful for showing that a set is countable DEF An uncountable set is NOT 1 1 with any countable set THM Cantor 1890 07 1 Q for the wheel of fortune is an uncountable setl Proof Suppose 07 1 is countable Index all a E 07 1 as 3617 2 Let acn have binary expansion acn 0307113071236713 where mnj 0 or 1 Let ynj 1 mm Then 3 0y11y22y33 y acn for all n Since 9 07 1 for the wheel of fortune is uncountable7 the third axiom of probability does not hold in the 0 1 77proof77

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

#### "When you're taking detailed notes and trying to help everyone else out in the class, it really helps you learn and understand the material...plus I made $280 on my first study guide!"

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.