Mathematics 220
Homework 8
1. Find the domain and range of the function f (x) =
must provide detailed proof).
Due Monday/Tuesday March 14/15
x1
x
(assume x is real). (Remember, you
2. Let f : R R be t
Mathematics 220
Practice Midterm
Page 1 of 5
This midterm has 4 questions on 5 pages
Read all the questions carefully before starting to work.
Give complete arguments and explanations for all your c
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, t
Mathematics 220
Homework for Week 4
Due January 31
There are 9 questions worth a total of 32. Not all questions will be graded though.
1. 2 marks 2.46
2. 3 marks Let P be the statement x R, y R s.t.
Mathematics 220
Homework Set 7
Due March 6
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 fro
Mathematics 220
Homework for Week 6
Due February 26
If you are using the 2nd edition, be careful question numbers may not agree.
4.28, 4.38
8.32, 8.38, 8.40, 8.42, 8.46, 8.50
EQ1 Let A be the set cf
Mathematics 220 Homework 2 - Solutions
1. (2.20) For statements P and Q, construct a truth table for (P Q) ( P ).
Solution:
P
T
T
F
F
Q
T
F
T
F
P Q (P Q) ( P )
T
F
F
T
T
T
T
T
P
F
F
T
T
2. (2.22) Cons
Mathematics 220
Homework 8
Due March 21
There are 11 questions worth a total of 26. The rst 6 questions are on material you
should know for the Test on March 14. Solutions for questions worth 0 point
Mathematics 220 Homework 6 - Solutions
1. (5.36) Let a, b R. Prove that if ab = 0, then a = 0 by using as many of the three proof
techniques as possible.
Solution:
Proof by contrapositive:
We prove th
Mathematics 220 Homework 3 - Solutions
1. (3.16) Let x Z. Prove that if 7x + 5 is odd, then x is even.
Solution: We will prove the contrapositive:
If x is odd, then 7x + 5 is even
Assume that x is odd
Mathematics 220
Homework for Week 3
Due January 30
Problems from Chapters 2 and 3 of the 3rd edition of the text.
2.48, 2.54, 2.60, 2.64, 2.68, 2.72
3.2, 3.8, 3.14, 3.18, 3.22
If you are using the 2
Mathematics 220 Homework 5 - Solutions
1. (4.28)
(a) Prove that if r is a real number such that 0 < r < 1, then
1
4.
r(1 r)
Solution:
Proof:
Assume that 0 < r < 1. Since (2r 1)2 0, it follows that
(2
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 = 2
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 abo
Mathematics 220
Practice Midterm March 9th 2013
Page 1 of 8
This midterm has 6 questions on 8 pages, for a total of 40 points.
Duration: 75 minutes
Read all the questions carefully before starting to
Mathematics 220 Homework 4 - Solutions
1. (3.18) Let n Z. Prove that (n + 1)2 1 is even if and only if n is even.
Solution: First, note that (n + 1)2 1 = n2 + 2n + 1 1 = n2 + 2n. We must prove the
two
Mathematics 220
Homework Set 5
Due: February 13
If you are using the 2nd edition, be careful question numbers may not agree.
4.32 (you might nd the result in EQ1 useful)
4.34 (the triangle inequalit
SOLUTVoMS
The University of British Columbia
Midterm Exan'l 1 - May 30, 2013
Mathematics 220 - section 921
Closed book examination Time: 1 hour
Last Name _ First _ Signature
Student Number
Special Ins
Mathematics 220
6 marks
Midterm 2 November 14th 2011
Page 1 of 3
1. (a) Let S R be a set. Dene precisely what it means for S to be well ordered.
Solution: S is well ordered if every non-empty subset o
SOLUTWNS
The University of British Columbia
Midterm Exam 2 ~ Jam 25: 2013
Matl'lex'naticss 220 -» Section 9521
Closed hook examination Time: 90 minutes
Last Name _ First mmmwmww Signature
Student Numb
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
Homework 8
Zoe Fox
17469157
November 3, 2016
1) C8Q12 - If A, B and C are sets, then A (B C) = (A B) (A C)
proof
To prove the proposition, we will prove A (B C) (A B) (A C)
and (A B) (A C) A (B C).
To
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 = 2
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. B
Mathematics 220
Homework for Week 3
Due: September 26, Friday
Problems from Chapters 2 of the 3rd edition of the text.
2.18, 2.20, 2.32, 2.48, 2.54, 2.60, 2.64, 2.68, 2.72
If you are using the 2nd ed
HW Set 1, Due: September 12, 2014, Friday
Page 30: 1.2, 1.4 (a)(d)(e), 1.8 (b)(e)
Page 31: 1.14, 1.16
Determine the cardinality of each of the following sets:
(1) A = cfw_0, cfw_0
(2) B = cfw_2, 3,
Mathematics 220
Homework for Week 4
This set will not be collected
Problems from Chapters 3 and 4 of the 3rd edition of the text.
3.2, 3.8, 3.14, 3.18, 3.22, 3.26
Prove the implication: if m, n Z ar
Mathematics 220
Homework for Week 2
Due September 19, 2014, Friday
Problems from Chapters 1 and 2 of the 3rd edition of the text.
1.22, 1.24, 1.38, 1.42, 1.54, 1.66, 1.74, 1.84
2.2, 2.6
If you are u
Mathematics 220
Homework 2 solutions
Solution:
1.22 The sets are
A [ B = cfw_1, 3, 5, 9, 13, 15
A \ B = cfw_9
A B = cfw_1, 5, 13
B A = cfw_3, 15
A = cfw_3, 7, 11, 15
A \ B = A B = cfw_1, 5, 13
1.24 Le
CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS
PART ONE
With the notion of bijection at hand, it is easy to formalize the idea that
two finite sets have the same number of elements: we just need to verif
Mathematics 220
Homework 2, Due: September 22
1. Section 1.5: #4 (a)-(h),
Solution:
We are given: A = cfw_b, c, d, B = cfw_a, b. Some simpler sets that appear in the
questions:
A B = cfw_(b, a), (b, b
Homework 8
1. (Chapter 10: Question 8) If n N, then
1
2
3
n
1
+ + + +
=1
.
2! 3! 4!
(n + 1)!
(n + 1)!
Proof. We will prove this by using induction on n.
Base step: When n = 1 the left hand side is
whi
Mathematics 220
Midterm Exam, 2:00pm
Page 1 of 7
This midterm has 6 questions on 7 pages Duration: 50 minutes
Read all the questions carefully before starting to work.
Give complete arguments and ex
Homework 1
Section 1.1, Q 12: In this question, we want to list the elements of the set
cfw_x Z : |2x| < 5.
First of all lets try to understand what this notation means. By the definition of the
set