Mathematics 220 Workshop 1
1. State and prove or disprove the negation of each of the following statements:
(a) k Z such that k odd and k 2 even.
(b) x, y R, x = y = x2 + y 2 > 0.
Solution: (a) The negation of the statement is:
k Z, k is even or k 2 is od
Mathematics 220
Solutions to Homework 5
5.4, 5.12, 5.16, 5.20, 5.28
5.24, 5.32, 5.36, 5.40
Let x, y be nonnegative numbers. Use a direct proof and a proof by contradiction
real
to show: If x < y, then x < y.
Solution:
5.4 We seek n N so that n(n + 1)/
16.6 (a) Given s> 0, let N= lkl/s.
% (b) Given 5) 0, let N= (1/5)
. Thenn > N implies n" > 1/5, so (l/n) < 5.
(c) 3".3: 5 s§,sogivm£>0,lctN=5/6.
n+2 n+2 n
(d) 110151,sogiv¢:nt:>0,letN=l/5.
H
i _ _ . . 2 .
2-3 %n n l 2 n _
So given £>0, le
Mathematics 220
Homework Set 10
Not collect
1. 12.4
2. 12.6
3. 12.8
4. 12.48
5. 12.10 (for (a) you may try induction)
6. 12.7 (the textbook has a solution to this question, but you should try to solve it
before reading the solution)
7. For each of the fol
Mathematics 220
Homework Set 9
Due: November 21
If you are using the 2nd edition, be careful question numbers may not agree.
10.20, 10.24
10.26, 10.28,
10.42 (draw a picture and think carefully about cases)
10.46 (induction is your friend)
EQ1 Let S,
Mathematics 220
Homework Set 10
Not collect
1. 12.4
Solution: We need |(n2 + 1)1 | < , so
n2 + 1 >
n2 >
1
Thus we pick N =
1
1
1
1 .
1
Proof. Let > 0 and pick N =
|(n2 + 1)1 0| =
1 . Now let n > N , then
n2
1
1
< 2
+1
N +1
Hence the sequence converges to
(:{171f'7/f41+<{4)L=%(/ZH)(V£H)7(+/)L
;/6(/zm (/Iz<z{2+1)+6(¢2+0): % Mum lm/2+ g)
: () (/f'l)(1k+}):/é(LH/(%,H).+Q (1(é+,)+y/
H Um /n w b} (MW 777
% q/K I w I f Y\ + I
W + 47%?! 7m) 472w)?! w! (WW)
w (mm/r) 1E}an 6") (2w) 31L,
5: (inln/ 7W): {2n+])(2,n+
Math 220 Homework 5
Due Friday, October 14
Content: 6.1, 6.2, 6.3, 7.1
Chapter 6:
#4, 8, 14, 18, 22
Chapter 7:
#6
Prove the following proposition:
Given any a Q and any c R,
c Q if and only if a + c Q.
Prove the following proposition:
For every a, b Z
MATH 220
Homework 3 (LaTeX)
Name: Qifan Tang
Student number: 32340151
Section 2.1
Decide whether or not the following are statements. In the case of a statement,
say if it is true or false, if possible.
10.(R N) (N R) = (N N)
It is a statement because it
MATH 220
Homework 2 (mostly written by LaTeX)
Name: Qifan Tang
Student number: 32340151
Section 1.5
4. Suppose A = cfw_b, c, dand B = cfw_a, b. Find
(a) (A B) (B B)
Obviously, the answer is cfw_(b, b), (b, a) for theyre the only common elements
between A
Homework 5
Qifan Tang
32340151
October 14, 2016
Written by LaTex
Chapter 6
Use the method of proof by contradiction to prove the following statements. (In each case, you should also
think about how a direct or contrapositive proof would work. You will fin
Homework 7
Qifan Tang
32340151
October 28, 2016
Written by LaTex
Prove the following statements.
1.If a Z, then 4 - (a2 3).
P roof.(By contradiction) Assume 4|(a2 3).
There must exist an integer k such that a2 3 = 4k.
It follows a2 = 4k + 3
Case 1. When a
Practice Midterm Exam Math 220
Problem 1. Use Mathematics Induction to show: Vn 2 4, n E N, n! 2 n2. (Recall: n! = 1-2-. 71)
4 {9 Wk =4, 4-! =4~3-z~l=2¥
H. 41= l6
4. > *2.
® Asswwt E k. zkl, TQM,
(k'H) .' = 0141) ' ; (It-H). k
3ch /L(I-I)7/I Pry kaga
k"
a
3.2 (a) True: comment in rst paragraph on page 18.
(b) False: its called a contradiction.
(0) True: comment after Practice 3.8.
(d) True: end of Example 3.1.
(e) False: must show p(n) is true for all n. /
3.4 (a) If all violets are blue, then all roses
Mathematics 220
Homework Set 8
If you are using the 2nd edition, be careful question numbers may not agree.
1. 9.4
Solution:
(a) R1 is a function from A1 to R.
(b) R2 is not a function from A2 to R. Both (9, 1) and (9, 1) are in R2 .
(c) R3 is not a funct
Mathematics 220
Homework Set 7
Due: October 31
If you are using the 2nd edition, be careful question numbers may not agree.
8.6, 8.12, 8.28, 8.32, 8.38, 8.40, 8.42, 8.46, 8.50
EQ1 Let A be the set cfw_1, 2, 3. Answer the following:
(a) Consider the rela
Mathematics 220 Workshop 1
1. State and prove or disprove the negation of each of the following statements:
(a) k Z such that k odd and k 2 even.
(b) x, y R, x = y = x2 + y 2 > 0.
Solution:
2. Determine whether each of the following is true or false, and
Mathematics 220 Workshop 3
1. Write down the following
(a) Denition of limn an = L
(b) Denition of a sequence cfw_an is bounded, an R.
(c) Denition of
n=1 an
converges.
2. Is the solution to each of the following question correct?
(a) For any x, y R with
Mathematics 220 Workshop 2
1. The proofs below contain errors. Please identify these errors, and prove or disprove the original
statement.
(a) Statement: x R such that for any y R, y 0, we have xy = 2x.
Proof: Let x R, and y = 2. Then, y 0, and xy = 2x, a
Mathematics 220 Workshop 2
1. The proofs below contain errors. Please identify these errors, and prove or disprove the original
statement.
(a) Statement: x R such that for any y R, y 0, we have xy = 2x.
Proof: Let x R, and y = 2. Then, y 0, and xy = 2x, a
‘ Problem 1 (15 points). Determine whether the following series converge or diverge. Justify your
answers.
C 0° ﬁ+sinn {Tn-1- 5M” > W‘i = _L
“QM—v / mm “ Problem 2 (15 points). Prove the following limits exist and then evaluate them.
(W _ ems)
(a) lim
Final Examination December 9th 2011
Mathematics 220
Page 1 of 13
This nal exam has 8 questions on 13 pages, for a total of 100 marks.
Duration: 2 hours 30 minutes
Full Name (including all middle names):
Student-No:
Signature:
UBC Rules governing examinati
FINAL EXAM
Math 220 Section 101
December 4, 2008
Last Name:
First Name:
Student Number:
Signature:
The exam is worth a total of 100 points with duration 2.5 hours. No books, notes or calculators
are allowed. Justify all answers, show all work and explain
Mathematics 220.101 Midterm Exam 1 Page 2 of 6
12 marks 1. (a) Is the following statement true or false? Justify your answer. Write its negation.
VT 6 R. Vy E R. 3: E R s.t. (:r < y) => (:r < 2 < y).
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Skamcn‘f 1.5 “’04- 73cca.u.s¢ For a.“ 1.;
00¢ CM CA