- May 15, 2020

MA多伦多大学 MAT 137课业解析题意：完成三道计算题解析：第三题：

. For which positive integers n ≥ 1 does 2^n > n^2 hold? Prove your claim by induction.

证明：n>=5（1）当 n=5 时，2^5=32 > 5^2=25，不等式成立 （2）假设 n=k (k>5)时，2^k > k^2; 则 n = k+1 时，2^(k+1)=22^k > 2(k^2)=(k-1)^2-2+(k+1)^2

当k>5时，(k-1)^2-2>0

所以 2^(k+1)>(k+1)^2

即 n>5 时，假设成立

由数学归纳法可知，V n>=5，2^n>n^2。涉及知识点：数学归纳法，集合更多可加微信讨论微信号：IT_51zuoyejunpdf

MAT 137Problem Set #1Due on Thursday September 26, 2019 by 11:59 pmSubmit via CrowdmarkInstructions• You will need to submit your solutions electronically. For instructions, see theMAT137 Crowdmark help page. Make sure you understand how to submit andthat you try the system ahead of time. If you leave it for the last minute and yourun into technical problems, you will be late. There are no extensions for any reason.• You will need to submit your answer to each question separately.• You may submit jointly written answers in groups of up to two people. Your partnercan be anyone in MAT137 from any lecture section. You can also submit jointlywritten answers with a different person for each problem set.• If you do not jointly write your solutions with someone else then you must submityour answers individually.• This problem set is about the introduction to logic, notation, quantifiers, conditionals, definitions, and proofs (Playlist 1).Problems0. Read Notes on Collaboration on the course website. Copy out the following sentenceand sign below it, to certify that you have read the \Notes on Collaboration”.\I have read and understood the notes on collaboration for this course, asexplained in the course website.”If submitting as a group of two, both people must sign and submit.1. Negate the following statement without using any negative words (\no”, \not”,\none”, \zero”, etc.):\All students at a university in Canada are enrolled in an odd-numbered coursethat is taught by a professor whose last name starts with a letter alphabeticallybefore Q and who lectures only on weekdays.”2. In this problem we will only consider (real-valued) functions with domain R. Wedefine two new concepts. Let f and g be two functions.• We say f is a rival of g if9x 2 R s.t. 8y 2 R; x < y =) jf(x) - g(x)j < jf(y) - g(y)j• We say f is a frenemy of g if8x 2 R; 9y 2 R s.t. x < y AND jf(x) - g(x)j < jf(y) - g(y)jBelow are four claims. Which ones are true and which ones are false? If a claim istrue, prove it. If a claim is false, show it with a counterexample.(a) If f and g are any two functions and f is a rival of g then f is a frenemy of g.(b) If f and g are any two functions and f is a frenemy of g then f is a rival of g.(c) If f and g are any two functions and f is a rival of g then g is a rival of f.(d) Let f; g, and h be any three functions. If f is a frenemy of g and g is a frenemyof h then f is a frenemy of h.3. For which positive integers n ≥ 1 does 2n > n2 hold? Prove your claim by induction.