# proofs

1. ### On the set N Ã— N, define the following relation: (a, b) âˆ¼ (c, d) if and only if a + d

On the set N Ã— N, define the following relation: (a, b) âˆ¼ (c, d) if and only if a + d = b + c. (1) Show that this is an equivalence relation (2) Describe the equivalence class of (1, 1)
2. ### Prove that (âˆ€n âˆˆ N)(âˆ€x âˆˆ Z)[x 2n is congruent to 0 or 1 modulo 4].

Prove that (âˆ€n âˆˆ N)(âˆ€x âˆˆ Z)[x^2n is congruent to 0 or 1 modulo 4]. and Using the previous problem to prove that âˆ’1 + 4x + x^2 + 8x^3 + x^4 = 0 has no integer solutions
3. ### Proofs -

One of DeMorganâ€™s laws (for logic), is that: Â¬(P âˆ§ Q) is logically equivalent to (Â¬P ) âˆ¨ (Â¬Q). Use this to prove DeMorganâ€™s first law for sets: If. A,B,C are sets, then A\(Bâˆ©C)=(A\B)âˆª(A\C).
4. ### Proofs

Create an example of a function f : R â†’ R such that f(f(f(R))) = f(f(R)) does not equal f(R). I donâ€™t even know how to approach this been struggling a lot
5. ### Logic proofs

If A is in Nairobi then A is in kenya. Outline a proof by contraposition of this theorem.
6. ### How to motivate students to do proofs?

I am finding it difficult to motivate students on why they should know how to prove mathematical results. They learn them just to pass examinations, but show no real interest or enthusiasm for this. How can I inspire them to love essential kind of mathematics? They love doing mathematical...
7. ### Academic Guidance Must I know Geometry Proofs?

Is knowledge of geometric proofs essential to success in Pre-Calc or Calc? I understand the basic reasoning behind two-column proofs used in geometry (especially when applied to algebra) but know virtually none of the geometry definitions or theorems. I also know that I do not enjoy studying...
8. ### God's Book of Proofs

A diverting article on an interesting book. https://www.wired.com/story/in-search-of-gods-mathematical-perfect-proofs/amp
9. ### Deeper Understanding of Limit Proofs

In an "N epsilon proof of sequence convergence", we tend to assume the limit. For example: If we were proving that the sequence 1/n as n goes from 1 to infinity, converges to 0, we would start with |1/nâˆ’0|<Ïµ. As you can see we initially assume that the limit is 0. However, what if we thought...
10. ### Intro to Math Proofs - Induction

Hello all, I am following the MIT OpenCourseware Mathematics for computer science. The current topic is proof by induction. In Lecture 2 at the ~1:00:00 mark, this problem is given: A 2^n x 2^n square can be covered by an L (nxn)shaped tile such that there is one open tile in the center. n n...
11. ### No. of math proofs created yearly

How many math proofs are created yearly? I recall the number is about 100,000.
12. ### Proofs

How can l prove that (a,b)=2(a/2,b/2) if a and b are positive integers

14. ### Introduction to reading proofs

I am currently working through a short introduction on reading proofs. https://math.berkeley.edu/~hutching/teach/proofs.pdf I am stuck on being able to read/understand the following quantifier and statement. (\forall x \in \mathbb{Z})(\exists y \in \mathbb{Z})x = 2y As I know, this statement...
15. ### Epsilon Delta Proofs

I don't really understand the goal of these. What is my objective? Is it to find a relationship between delta and epsilon? Could someone please do out this example to illustrate the proper procedure and purpose of these proofs? lim as x approaches 2 of x^3 is 8.
16. ### LOGIC Integral set proofs?

So I've got all but like two problems on my assignment for this week figured out. https://gyazo.com/adc99eed1f3ca937608512d7bb015b5f sorry for not just typing out my problem I'm not sure how to use math script in text so I figured this would be faster. I would like to request that...
17. ### ï¼ï¼ï¼Need help for Proofs including rank-nullity formula

hi guys can you help me in solving this question? T: R^3 -> R^3 is a linear transformation. Prove the equivalence of the following ( R^3 =ker+im if for all v belongs to R^3, there exists x belongs to ker(T) and y belongs to im(T) such that v =x+y and ker(T)intersects im(T)= 0 ) a) R = ker(T)...
18. ### Is there beauty in mathematical proofs?

Hi, I've been put to answer this question as a research report and I was wondering if any of you had any ideas of certain proofs/topics which I should look into to show their beauty or even proofs which you feel are not beautiful and if you could say why? Also, do you feel beauty in a proof...
19. ### Limit proofs

I have to limits which I wish to prove 1. \lim_{x\to \infty} \frac{e^x}{x^n} = \infty. Prove that for any positive integer n. The exponential function approaches infinity faster than any power of x. 2. \lim_{x\to \infty} \frac{lnx}{x^p} = 0. Prove that for any number p > 0. The...
20. ### How can I better understand proofs

When I learn proofs for something, I tend to be unable to remember all of the required information, i.e. what all of the variables mean and what conditions they have. Furthermore, I often don't know how one statement follows from another and don't know how to find out. So, how can I better...