计算机科学的数学表达与推理代写 CSC 165 H1代写 数学代写

CSC 165 H1

Term Test 3 — Question 1 of 4

计算机科学的数学表达与推理代写 Aids Allowed: Your own notes taken during lectures and office hours, the lecture slides and recordings (for all sections), and the Course

Aids Allowed: Your own notes taken during lectures and office hours, the lecture slides and recordings (for all sections), and the Course Notes (textbook).

Submission Instructions 计算机科学的数学表达与推理代写

• Submit your work directly on MarkUs—even if you are late!

• You may type your answers or hand-write them legibly, on paper or using a tablet and stylus.

• You may write your answers directly on the question paper, or on another piece of paper/document.

• You may submit your answers as a single file/document or as multiple files/documents. Each document may contain answers for only part of one question, an entire question, or multiple questions, but please

label each part of your answers to make it clear what you are answering.

• You may name your file(s) any way you want (there is no “required file”). 计算机科学的数学表达与推理代写

• You must submit your answers in PDF or as photos (JPEG/JPG/GIF/PNG/HEIC/HEIF). Other formats (e.g., Word documents, LATEX source files, ZIP files) are NOT accepted—you must export or compile documents to PDF, convert images into a supported format, and upload each file individually.

For all questions in this test, “proof” means a formal proof that includes a header, and a proof body with justifications for each deduction. Each question can be answered correctly in less than one (1) page. You will NOT be penalized directly if you use more space for your answer, but longer answers increase the chance of errors… Remember that we are looking for evidence that you understand the conventions for writing correct proofs, so pay attention to the structure of your answers in addition to their content!

1. [5 marks] Asymptotic Notation I. 计算机科学的数学表达与推理代写

You may use https://www.desmos.com/calculator to look at the graph of the function in this question, but NO other online resource is allowed for any question on this test. Also, you still need to provide rigorous arguments for each proof: remember that a graph is NOT a rigorous argument.

Consider the function f : N → R^≥0 defined by the formula

f(n) = n²(cos(nπ) + 1) + 1.

Prove that:

(a) [2 marks] f ∈ O(n²)

(b) [3 marks] f ∉ Θ(n²)

2. [3 marks] Number Representations. 计算机科学的数学表达与推理代写

Write the following natural numbers x. Feel free to write them as sums. No proof is required for this question!

(a) The largest number x such that (x)₂ is 4-digits long.

(b) The smallest number x such that (x)₁₆ is 5-digits long and contains exactly two A’s and one E, with no leading 0’s.

(c) The smallest number x such that (x)₈ is a 5-digit long palindrome, by which we mean a number that reads the same forward and it does backward, i.e., 737 or 24542, with no leading 0’s, where each digit appears at most twice.

3. [4 marks] Induction.

Warning! This question does not require deep insight but it is longer to write up (you may need more than 1 page). You should keep it for last. Also, you will receive at most half the marks if you do NOT use induction.

Let a₀, a₁, a₂, . . . ∈ R and b₀, b₁, b₂, . . . ∈ R be arbitrary. Prove the following statement by induction: for all n ∈ N, if n ≥ 2, then

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