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MATH1061 Assignment 3

 Due 1pm Thursday 7 Oct 2021 This Assignment is compulsory, and contributes 5% 

towards your final grade. It should be submitted by 1pm on Thursday  

In the absence of a medical certificate or other valid documented excuse, assignments submitted after the due

 date will not be marked. Prepare your assignment as a pdf file, either by typing it, writing on a tablet or by

 scanning/photographing your handwritten work. Ensure that your name, student number 

and tutorial group number appear on the first page of your submission. Check that your pdf file is legible and

 that the file size is not excessive. Files that are poorly scanned and/or illegible may not be marked.

 Upload your submission using the assignment submission link in Blackboard. Please note that our online 

systems struggle with filenames that contain foreign characters (e.g. Chinese, Japanese, Arabic) so please

 ensure that your filename does not contain such characters. 1. (6 marks) Recall that a sequence a0, a1, . . . is a 

geometric sequence if and only if there exists a constant r such that ak = r · ak−1 for each integer k ≥ 1.

 An explicit formula for this geometric sequence is an = a0 · rn for each integer n ≥ 0 and the sum of the

 first n + 1 consecutive terms of this geometric sequence is given by n∑ i=0 ai = a0(1− rn+1) 1− r . 

Consider the sequence defined recursively as a1 = 1, ak = k−1∑ i=1 ai for k ≥ 2. (a) Calculate the values of a2, a3, a4 and a5. (b) Guess an explicit formula for this sequence and prove that your guess is correct. 2. (12 marks) Prove or disprove each of the following statements. (a) For all sets A and B, P(A)× P(B) = P(A×B). (b) For all non-empty sets A, B and C, if A×B = A× C, then B = C. (c) For all sets A, B and C, A× (B − C) = (A×B)− (A× C). 3. (4 marks) Define a function f : P(Z+)→ Z≥0 where f(A) = { 0 if A = ∅ the least element of A otherwise. (a) Is the function f injective? Justify your answer. (b) Is the function f surjective? Justify your answer. 4. (8 marks) Let S be the set of all rational numbers between 0 and 1, that is, S = {x ∈ Q : 0 < x < 1}. Use the Schro¨der-Bernstein Theorem to prove that |S| = |Z+|.