### Create a StudySoup account

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

Already have a StudySoup account? Login here

# Math 3040 Prove It! Week 1 3040

Cornell

### View Full Document

## About this Document

## 63

## 2

## Popular in Prove It!

## Popular in Mathematics (M)

This 5 page Class Notes was uploaded by Anudeep Gavini on Tuesday January 26, 2016. The Class Notes belongs to 3040 at Cornell University taught by Edward B Swartz in Winter 2016. Since its upload, it has received 63 views. For similar materials see Prove It! in Mathematics (M) at Cornell University.

## Similar to 3040 at Cornell

## Reviews for Math 3040 Prove It! Week 1

### 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: 01/26/16

Math 3040 - 1/27/16 February 7, 2016 1 Syllabus Notes ▯ There will be 1 in class prelim, some time near the drop deadline. ▯ There will be 1 in class ▯nal, date to be determined. ▯ There will be 1 paper, due near the end of the semester, to be writen in LaTeX. ▯ Finally, there will be up to 2 retries for all homeworks turned in, for a maximum of 100 2 How is Math di▯erent from other disciplines? ▯ highly theoretical ▯ Mostly inductive, whereas other disciplines require more deduction. { However, discovering a new property in mathematics still requires some deduction ▯ Math permeates all sciences, whereas not all sciences permeate the other sciences ▯ Math is precise enough to rely only on formal logical proofs to derive ideas ▯ Once something is proven true, it remains that way forever. { it is very normal for math papers to have 25 year old references. Conversely, this is rare in scienti▯c disciplines, as hypotheses are often adjusted or disproved. { There exists some debate on what constitutes a valid proof, however - for example, some mathematicians don’t accept proof by contradic- tion ▯ Mathematics is less tangible - exists more conceptually 1 ▯ Math is a vehicle for other disciplines ▯ A question has one answer, but multiple paths to get there. ▯ In▯nities are an important concept in math, whereas other disciplines tend to deal with the ▯nite. 2 Math 3040 - 1/29/16 February 7, 2016 1 Sample Proof 1 Proposition: 4(1 ▯ 1+ 1▯ :::) = ▯ 3 5 7 Proof: d (arctanx) = 1 dt 1 + x2 1 = 1 ▯ (▯x ) = 1 + (▯x ) + (▯x ) + (▯x ) ::: = 1 ▯ x + x ▯ x ::: R By the fundamental theorem of calculus, x 12dt = arctan(x) ▯ arctan(0) = 0 1+t arctan(x). Thus, we can write: Z x t3 t5 t7 1 ▯ t + t ▯ t + t :::dt = [t ▯ + ▯ ::0] 0 3 5 7 3 5 7 x x x = x ▯ + ▯ ::: 3 5 7 = arctan(x) 1 1 1 1 ▯ Plugging in x = 1 gives 1 ▯ 3+ 5 ▯ 7 + 9::: =4 . Thus, ▯ = 4arctan(1) = 4(1 ▯ 1 + 1 ▯ ::::) 3 5 7 ▯ Well, this proof is ▯ne, but there are some things it cannot answer some questions. ▯ For example, how can we ▯nd the ▯rst 3 digits of pi using this proof? How many terms will we need to calculate to ▯nd out the answer? 1 ▯ Also consider the following proposition: Take any x 2 R. Then, there 1 1 exists a reordering of the terms 1;3; 5::: expressed as a1;a 2a 3:: such that ▯1a i x. This requires a deep understanding of the sample proof! 2 Integers ▯ There exist 2 approaches to ▯nd the properties of a set Z { We can try and derive each and every property, which is very di▯cult { Or, we can make some assumptions, called axioms, that we don’t need to prove ▯ We will take the Axiomatic approach. The axioms of the the integers are as follows: { Axiom 1.1: ▯ m + n = n + m (Commutativity of addition) ▯ (m + n) + p = m + (n + p) (Associativity of addition) ▯ m ▯ (n + p) = (m ▯ n) + (m ▯ p) (Distribution) ▯ m ▯ n = n ▯ m (Commutativity of multiplication) ▯ (m ▯ n) ▯ p = m ▯ (n ▯ p) (Associativity of Multiplication) { Axiom 1.2: There exists an element 0 2 Z such that for all n 2 Z, n + 0 = n. This is the Additive Identity. { Axiom 1.3: There exists an element 1 6= 0;1 2 Z such that for all m 2 Z, m ▯ 1 = m. This is the multiplicative identity. { Axiom 1.4: For all elements m 2 Z there exists a value ▯n such that n + ▯n = 0. This is the Additive inverse. { Axiom 1.5: Suppose. m;n;p 2 Z,and m = 6 0. If m ▯ n = m ▯ p, then n = p. This is Cancellation. Here is a sample proof using these axioms: 3 Sample Proof 2 Prop 1.11 v): Prove m(n + (p + q)) = (mn + mp) + mq. Proof: m(n + (p + q)) = m((n + p) + q) By associativity = m(n + p) + mq By distribution = (mn + mp) + mq By distribution Notice that we wrote out information for each and every step. Do we really need to write so much? It would be tedious for larger proofs. 2 However, once you show that you know what you’re doing, you can always combine multiple small steps together, such as combining associativity and com- mutativity in one step. 4 What’s wrong with this proof? Prop: (▯m)(▯n) = mn Proof: First, add (▯m ▯ n) to both sides. This gives (▯m)(▯n) + (▯m)n = mn + (▯m)n. By distributivity, we can reduce this to ▯m(▯n + n) = n(m + ▯m). By axiom 1.4, this reduces to ▯m ▯ 0 = n ▯ 0. By a previous proposition, this gives 0 = 0. ▯ This doesn’t actually prove anything, it just reduces a statement to a known fact. ▯ This is NOT the same as proving that the left side is, in fact, equal to the right side. ▯ in the case of this proof, you should use the axioms and previously proved statements to show that the left equals the right. 3

### 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."

#### "I signed up to be an Elite Notetaker with 2 of my sorority sisters this semester. We just posted our notes weekly and were each making over $600 per month. I LOVE StudySoup!"

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "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.