MATH1004代写、代写Java,c++编程

日期：2022-08-25 01:52

The University of Sydney

School of Mathematics and Statistics

Assignment 1

MATH1004: Discrete Mathematics Semester 2, 2022

Lecturer: Oded Yacobi

This individual assignment is due by 11:59pm Thursday 25 August 2022, via

Canvas. Late assignments will receive a penalty of 5% per day until the closing date.

A single PDF copy of your answers must be uploaded in the Learning Management

System (Canvas) at https://canvas.sydney.edu.au/courses/44703. Please sub-

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working on problems, but students must write up and submit their own version of the

solutions. If you have technical difficulties with your submission, see the University

of Sydney Canvas Guide, available from the Help section of Canvas.

This assignment is worth 5% of your final assessment for this course. Your answers should be

well written, neat, thoughtful, mathematically concise, and a pleasure to read. Please cite any

resources used and show all working. Present your arguments clearly using words of explanation

and diagrams where relevant. After all, mathematics is about communicating your ideas. This

is a worthwhile skill which takes time and effort to master. The marker will give you feedback

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Copyright ? 2022 The University of Sydney 1

1. Suppose you begin on a first rung of a ladder. A ladder path is a sequence of steps on

the ladder where each step is either up one rung or down one rung, you always remain

on the ladder, and you end back on the first rung. Assume the ladder is tall enough so

that you can never step off the top. For example, there is only one ladder path with two

steps, given by taking one step up and then one step down.

(a) Make a list of all the ladder paths with n steps, where n ∈ {3, 4, 5, 6}.

(b) Let n ≥ 1. Write down a correspondence that relates the ladder paths with 2n

steps to Catalan paths from (0, 0) to (n, n). More precisely, let Ln be the set of

ladder paths with 2n steps, and let Cn be the set of Catalan paths from (0, 0) to

(n, n). Construct a bijection f : Ln → Cn. (You don’t have to prove that it is a

bijection.)

2. (a) Find explicit bijections, and use the horizontal line tests to prove that they are

indeed bijections.

(i) f : (a, b) → (c, d), where a, b, c, d are real numbers such that a < b and

c < d.

(ii) g : (0, 1)→ (1,∞).

(b) For a set X, let

(

X

k

)

denote the set of subsets of X of cardinality k. Given n ≥ 1

we define three sets of cardinality n: Xn = {1, . . . , n}, X ′n = {1′, . . . , n′} and

X ′′n = {1′′, . . . , n′′}. For n > 3 construct a bijection