21-127 Concepts of Mathematics, Spring 2014
Homework 1
Due: Thursday, January 23rd
The following problems from the course textbook are assigned but will not
be collected:
Pages 48-49
Pages 83-86
:
:
1.3.6, 1.3.8, 1.3.10
1.5.7, 1.5.14, 1.5.19
The ve proble
Practice List Solutions
March 20, 2014
1. [Divisibility by 3] Prove that a number, x, is divisible by 3 i the sum of its digits
is divisible by 3. Start by assuming that x is a number with r + 1 digits as follows:
ar ar1 ar2 . . . a1 a0 .
Solution:
This i
Practice List Solutions
March 20, 2014
1. [Divisibility by 3] Prove that a number, x, is divisible by 3 i the sum of its digits
is divisible by 3. Start by assuming that x is a number with r + 1 digits as follows:
ar ar1 ar2 . . . a1 a0 .
Solution:
This i
21-127 Concepts of Mathematics
Spring 2014
Test 3 Solutions
Andrew ID:
Name:
Please circle your section:
Professors Picollelli and Harchol-Balters Sections:
Section A
12:30
Clive Newstead
Section B
1:30
Reilly Miller
Section F
10:30
Clive Newstead
Section
21-127 Concepts of Mathematics
Spring 2014
Test 1 Solutions
Andrew ID:
Name: Solutions
Please circle your section:
Professors Picollelli and Harchol-Balters Sections:
Section A
12:30
Clive Newstead
Section B
1:30
Reilly Miller
Section F
10:30
Clive Newste
21-127 Concepts of Mathematics, Spring 2014, MIDTERM II
SOLUTIONS
Name:
Andrew ID:
Please circle your section:
Professors Picollelli and Harchol-Balters Sections:
Section A
12:30
Clive Newstead
Section B
1:30
Reilly Miller
Section F
10:30
Clive Newstead
S
21-127 Concepts of Mathematics, Spring 2014, MIDTERM II
SOLUTIONS
Name:
Andrew ID:
Please circle your section:
Professors Picollelli and Harchol-Balters Sections:
Section A
12:30
Clive Newstead
Section B
1:30
Reilly Miller
Section F
10:30
Clive Newstead
S
21-127 Concepts of Mathematics, Spring 2014
Homework 10 Solutions
1. Proof: We prove this by strong induction on n. Well allow for two base cases: n = 1
and n = 2.
BC: If n = 1, the assertion is that |A1 | = |A1 |, which is true. If n = 2, this assertion
21-127 Concepts of Mathematics, Spring 2014
Homework 7
Due: Thursday, March 20th
The six problems given below will be collected:
DIRECTIONS: For each problem be very careful to show all your work and in particular
to cite every theorem/lemma/proposition/a
21-127 Concepts of Mathematics, Spring 2014
Homework 6
Due: Thursday, March 6th
The six problems given below will be collected:
DIRECTIONS: For each problem be very careful to show all your work and in particular to
cite every theorem/lemma/proposition/al
21-127 Concepts of Mathematics, Spring 2014
Homework 5 Solutions
1. Suppose xn Z for every n N, and that xn = 2xn1 + 3xn2 for all n 3.
a. Prove that if x1 and x2 are both odd, then xn is odd for all n N.
b. Prove that if x1 = x2 = 1, then for all n N,
xn
21-127 Concepts of Mathematics, Spring 2014
Homework 5
Due: Thursday, February 27th
The following problems are assigned but will not be collected:
1. (i) Page 136 : 2.7.15: 4. and 5.
2. (ii) Pages 358-362 : 5.7.10, 5.7.14 (this problem is challenging), 5.
21-127 Concepts of Mathematics, Spring 2014
Homework 4
Due: Thursday, February 13th
The following problems from the course textbook are assigned but will not
be collected:
Pages 134-137
Pages 357-359
:
:
2.7.1, 2.7.12, 2.1.13, 2.7.15:1.-3.
5.7.2, 5.7.5, 5
21-127 Concepts of Mathematics, Spring 2014
Homework 3 Solutions
Solutions to the Collected Problems:
1. a. Negation: x Z . P (x) Q(x).
The original statement is False, since if x = 1, then x Z, P (1) is True as 1 1 3,
and Q(1) is True since 1 is odd.
b.
21-127 Concepts of Mathematics, Spring 2014
Homework 3
Due: Thursday, February 6th
The following problems from the course textbook are assigned but will not
be collected:
Pages 303-307
:
4.11.2, 4.11.5, 4.11.6, 4.11.15, 4.11.16, 4.11.18, 4.11.20
The ve pr