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 the function defined by f (x) = x2 + 3x + 4.
(a) Prove 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 calculations; answers without
justications will not be m
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)/
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) Consider the statements:
P :
2 is rational ,
Q:
2
is ration
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. y 2 = x.
(a) State this in words (no notation allowed).
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 from A2 to R. Both (9, 1) and (9, 1) are in R2 .
(c) R3 is
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 points will be
posted by March 9.
1. 6 marks (a) Give an exa
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 cfw_1, 2, 3. Answer the following:
(a) Consider the relat
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. Then x = 2n + 1 for some integer n. Therefore,
7x + 5
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 2nd edition, be careful question numbers may not agree.
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 the contrapositive of the statement, which is: If a = 0 t
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
(2r 1)2 = 4r2 4r + 1 0.
Thus, 1 4r 4r2 = 4r(1 r). Since 0
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 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
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 for Week 2 Due January 19/20 in your section
There are 10 questions worth a total of 40.
1. 4 marks Let A = cfw_1, 2, . . . , 12. Give an example of a partition S of A satisfying the
following
requirements: (i) |S| = 5, (ii) ther
Mathematics 220 Midterm Exam 2, Section 101 Page 2 of 6
1. (a) Dene precisely what it means for a nonempty set S of the real numbers to be well
ordered.
A none.th 5 S 4 "0'! numbers :5 Salal fa be
wellordercal {P cam-.7 amen/1] salon} cf 5 has
a lens} e
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
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 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
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
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 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
Homework Set 8
Due November 7
If you are using the 2nd edition, be careful question numbers may not agree.
9.4, 9.6 (c)(e)
9.8. Then repeat Problem 9.8 but with B = cfw_10, 7.
9.12 (a) (c) (f), 9.16
9.20, 9.24, 9.26
9.56, 9.64, 9.70,
Mathematics 220
Homework Set 6
Due: October 24th
If you are using the 2nd edition, be careful question numbers may not agree.
6.2, 6.8, 6.12, 6.20(a)
EQ1. Show that for every integer n 2
1
1
22
1
1
n2
=
n+1
.
2n
EQ2. Show 5|(9n 4n ) for n N.
EQ3. Sh
Mathematics 220
Homework for Week 5
Due: October 17, Friday
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.
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 edition, be careful question numbers may not agree.
Dete
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, cfw_4, 5, cfw_2, 3, 4, 5
1
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 are odd then m2 + 3n2 is even. What is the converse
of th