York University
Math 1019
Final Exam
April 10, 2012
Last Name:
First Name:
Student #:
Question Points Marks
1
10
2
8
3
12
4
14
5
8
6
8
7
10
8
10
9
10
9
10
Notes. 1. The duration of the test is 180 minutes sharp. When the test is over, stop writing
instant
MATH/CSE 1019N Test 2 (Winter 2014)
Last Name:
First Name:
Student Number:
The test is 80 minutes long
No aides are permitted
In the questions below (1 point each) determine whether the set is finite or infinite. If the set is
finite, find its size.
x x Z
Math 1019, test 2 solutions retake (March 28) Section N, W13
[Note that I have added many more clarications and details than was required in the test, and therefore
the solutions given here are much longer than necessary.]
1 3
x + 2x2 , f2 (x) = 42 x, f3
Math 1019, test 2 retake (March 28) solutions, Section M W13
[Note that I have added many more clarications and details than was required in the test, and
therefore the solutions given here are much longer than necessary.]
[Comments in square brackets (li
Math 1019, test 2 solutions, Section M W13
1 3
x + 2x2 , f2 (x) = 42 x, f3 (x) = x3 + 2x2 , f4 (x) = 2x , f5 (x) = 1010 x,
(1) Let f1 (x) =
100
x3 + 10x2
x2 + x
, and f7 (x) = 4
f6 (x) =
.
3x 2
x + 10x3
Find out for which i and j we have that fi is O(fj )
MATH/CSE 1019N Test 1 (Winter 2014)
Last Name:
First Name:
Student Number:
The test is 80 minutes long
No aides are permitted
For questions 1- 17 answer TRUE or FALSE only (no partial credit!). You get 1
point for the correct answer, 0 points otherwise.
1
Math 1019, sections M and N, Fall 2013. Assignment #2 solutions.
(1) Prove that if x2 is irrational then x is irrational.
Solution. We use the method of indirect proof. Assume x is rational. Then x = a/b for some integers
a, b. So x2 = a2 /b2 ; since a2 a
Math 1019 W13, Assignment #4 solutions.
(1) Assume that the statement P (n) is true for n = 1 and that P (n) P (2n) for all n N.
Furthermore assume that if n > 0 then P (n) P (n 1). Prove (n N)P (n).
Solution. Let Q(n) be the statement that P (2n ) is tru