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MATH1061 Assignment
All assignments in this course must be submitted electronically and SUBMITTED AS A SINGLE PDF FILE.
Prepare your assignment solutions using Word, LaTeX, Windows Journal, or other application, ensuring that
your name, student number and tutorial group number appear clearly at the top to the first page, and then
save your file in pdf format. Alternatively, you may handwrite your solutions and scan or photograph your
handwritten work to create a pdf file. Make sure that your pdf file is legible and that the file size is not
excessive. Use the assignment submission link in Blackboard to submit the pdf file.
1. (6 marks) Consider the sequence {an}n1 defined recursively by
a1 = 2 and ak = 3ak1 + 2 for each integer 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. (4 marks) Prove the following statement using set identities. For all sets A,B and C, that are
subsets of a universal set U ,
A (B [ C) = (A B) C.
Show your work by naming the identities you use.
3. (4 marks) Define sets S, T and E as follows:
S = {n 2 Z | n ⌘ 0 (mod 4)}
T = {n 2 Z | n ⌘ 2 (mod 4)}
E = {n 2 Z | n ⌘ 0 (mod 2)}.
Prove that S ⇥ T ⇢ E ⇥ E.
4. (4 marks) Define a function f : P({0, 1})⇥ P({1, 2})! {0, 1, 2} by f((A,B)) =| A \B |.
(a) Is the function f injective? Justify your answer.
(b) Is the function f surjective? Justify your answer.
5. (6 marks) Let A and B be the following intervals of real numbers: A = [0, 2] and B = (2, 6).
Use the Schro¨der-Bernstein Theorem to prove that |A| = |B|.
