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 ne
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
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
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
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 ,
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 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 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
Solutions to Homework Set 6
6.2, 6.8, 6.12, 6.20
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. Show that for every integer n 4,
Mathematics 220
Solutions to Homework 4
3.2. Let n N. Prove that if |n 1| + |n + 1| 1, then |n2 1| 4.
Solution:
Result. Let n N. If |n 1| + |n + 1| 1, then |n2 1| 4.
Proof. Suppose that n N. Then n 1,
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 suc
Mathematics 220
Practice Midterm
Page 1 of 10
This midterm has 9 questions on 10 pages
Read all the questions carefully before starting to work.
Give complete arguments and explanations for all your
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'
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 al
(:{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)
Mathematics 220
Solution to Homework 3
2.18. (a) We need both P and Q to be true. So A can contain 1, 3, 5 but cannot be empty.
Hence A P (cfw_1, 3, 5) cfw_.
(b) We need that either P is true or Q is
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
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
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. Determi
Mathematics 220
Homework for Week 6 Due Febuary 27/28 in your section
1. Prove the following statement using (i) Proof by contrapositive (ii) Proof by contradiction: Let a, b Z. If a is odd and a + b
Mathematics 220
Homework for Week 5
1. 5.4 : Disprove the statement: Let n N. If
n(n+1)
2
is odd, then
Due Feb 9/10
(n+1)(n+2)
2
is odd.
2. 5.11: Prove that there is no smallest positive irrational nu
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. Let n N. Prove that if |n 1| + |n + 1| 1, then |n2 1| 4.
3.8. Pro
Mathematics 220
Practice Midterm 2
Page 1 of 6
This midterm has 5 questions on 6 pages
Read all the questions carefully before starting to work.
Give complete arguments and explanations for all your
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