MAST20026 Assignment 1
Semester 1 2022
This assignment consists of 2 pages.
Due Date: Friday March 18
Assignments are due in Gradescope on the due date listed above. Please see the instructions on the page where
you accessed this file for details on submitting your assignment. To ease submission, please prepare assignment
solutions for each question on its own page. To enable to graders to grade anonymously, do not include your
name anywhere on your assignment.
Students are encouraged to work together on understanding the problems and their solutions. However, submitted
solutions must be prepared individually, in your own words, and without the aid of others. For example, if you
are asked to give an example as part of your solution, your example should be meaningfully different from anyone
you have worked with on understanding the problems. Answers presented without justification will receive no
marks.
If you do not completely understand your submitted solutions, it is possible you are committing academic
misconduct. This is a serious offense.
Please do not upload this assignment paper or any of its questions to any online “help” service (e.g., CourseHero,
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more than likely violating the University’s academic misconduct policy.
PART A
To goal of assessments in this course is to help you know how well you are mastering the concepts presented in
lecture and tutorials. The work that you hand in for assessments is meant to reflect your level of understanding.
And so it is important that the work you hand in fully demonstrates your mastery of subject concepts.
Download the file called Guidelines for Good Mathematical Writing from the same place you accessed this file.
Read it and
write a short response (≤ 250 words) that considers the following questions:
Was there anything about the reading that you found surprising
What other experiences, if any, do you have in mathematics classes where you were expected to write in
full sentences as part of your solutions Do you find it difficult to write in mathematics courses
How will your approach to crafting your submission for this assignment change, if at all, as a result of
having done this reading
Note: In the reading above, the author uses the terms formal and informal differently than how we are using
them in our subject materials.
5 marks
PART B
(2) (a) (4 marks) For each of the implications below, write the contrapositive and the converse.
(i) If n = 41, then n is composite.
(ii) If x′ > 2r 4, then 2x′ 2 < r.
(b) (4 marks) Let p(x) : x > 2 over the domain R.
(i) If one exists, give an example of a condition q(x) for which p(x) = q(x) is true for every
x ∈ R, but q(x) = p(x) is not true for every x ∈ R. If no such example exists, briefly
explain why not.
(ii) If one exists, give an example of a condition q(x) for which p(x) = q(x) is true for every
x ∈ R and q(x) = p(x) is true for every x ∈ R. If no such example exists, briefly explain
why not.
1
(3) Consider the following theorem and its partially completed proof.
Theorem. There are infinitely many prime numbers.
Proof.
i. Let p be the largest prime number. (premise)
ii. Let p1, p2, . . . , pk be the list of prime numbers that are less than p. (i)
iii. Let n = p1p2 · · · pkp+ 1. (notation)
iv. n > p (iii)
v. n is not prime. ( )
vi. None of p1, p2, . . . , pk, p divide n. (iii, algebra)
vii. There exists a prime q ∈ [p+ 1, n 1] that is a divisor of n. ( )
viii. q > p. (vii)
ix. p is not the largest prime number. (viii)
x. ( ). (i,ix)