Thursday, December 19, 2013
Thursday, December 12, 2013
question answer key
| |||
I have a question on the answer key:
For
the recurrence relation problem (#5), isn't the characteristic equation
supposed to be r^2+2r+1=0 because of the negatives in the original
recurrence relation?
***********************
Yes, my bad. This would change the answer to -n(-1)^n.
Wednesday, December 11, 2013
Practice Test
The practice test may be accessed at this link.
The practice test answers may be accessed at this link.
The practice test answers may be accessed at this link.
Monday, December 9, 2013
Saturday, December 7, 2013
Tuesday, November 26, 2013
Homework due next week
Sections 6.4,8.1, 8.2,8.5, Due Thursday 12/5/2013
Also, I extended the due date for the webwork sections 8.2 and 8.5 to November 30.
Also, I extended the due date for the webwork sections 8.2 and 8.5 to November 30.
webwork question
I am not entirely sure what pairwise disjoint means in the problem
Find the number of elements in A1∪A2∪A3 if there are **** elements in A1, **** elements in A2, and **** elements in A3 in each of the following situations:
(a) The sets are pairwise disjoint.
(b) There are ** elements common to each pair of sets and * elements in all three sets
*************************************************
*************************************************
Pairwise disjoint means that that any pair of distinct sets have an intersection that is empty
Tuesday, November 19, 2013
Wednesday, November 13, 2013
Office hours cancelled
Office hours are cancelled today Nov 13 and Friday Nov 15. I have some medical issues to deal with and these were the times I could get. I'll be available Thursday afternoon if you need to speak to me.
Tuesday, November 12, 2013
Homework Due Next Week 11/18/2013-11/22/2013
1) Webwork sections 6.2 and 6.3 due Tuesday 11/19/2013 at 11:59pm
2) Book homework sections 6.2 and 6.3 due Thursday 11/21/2013 at the *beginning* of class
2) Book homework sections 6.2 and 6.3 due Thursday 11/21/2013 at the *beginning* of class
Thursday, November 7, 2013
Wednesday, November 6, 2013
Question on Euclid's proof of the infinitude of primes
By the fundamental theorem of arithmetic, Q is prime or else it can be written as the product of
two or more primes. However, none of the primes pj divides Q, for if pj | Q, then pj divides
Q −p1p2 ···pn = 1. Hence, there is a prime not in the list p1, p2,...,pn. This prime is
either Q, if it is prime, or a prime factor of Q. This is a contradiction because we assumed that
we have listed all the primes. Consequently, there are infinitely many primes.
This is the proof provided in our textbook (Rosen) that demonstrates the infinitude of primes. I find myself unable to grasp this proof.
Specifically, when it is stated that "if pj | Q, then pj divides Q −p1p2 ···pn = 1." I don't understand why the first proposition implies the second. Can you help me with this?
**************************************************************************
....judging from what you didn't put in your quote from the book, you may have missed the part that sets up the proof by contradiction: Suppose that there are a finite number of primes; write them as p_1, p_2, ... p_n. Define the integer Q as Q=p_1p_2...p_n+1. Here's where what you quoted comes in:
By the fundamental theorem of arithmetic, Q is prime or else it can be written as the product of
two or more primes. However, none of the primes pj divides Q, for if pj | Q, then pj divides
Q −p1p2 ···pn = 1. Hence, there is a prime not in the list p1, p2,...,pn.
Really, you can see from the definition of Q that Q div p_j is the product of all of the p's except p_j, and the remainder Q mod p_j is 1. This means that Q is not divisible by any of my finite list of primes, so it's not a composite number (i.e. a product of more than one prime), so it must be prime. But Q is also strictly bigger than any of my list of primes p_1, p_2, ..., p_n, so it is not one of p_1, p_2, ..., p_n. But these were supposed to be all of the primes. This is a contradiction. Therefore my supposition that there are only a finite number of primes is false. Therefore there are an infinite number of primes.
Monday, November 4, 2013
webwork question
Had issues with Section4.1 Rosen7: Problem 5 as practice indicates that
you are rounding up (ceiling) but this is not the case on the actual
homework problem. The homework problem did not like decimal spots, number
R number, or number plus fraction.
Please let me know how this problem was suppose to be answered.
Last answer:
AnSwEr0001: 2
AnSwEr0002: 0
AnSwEr0003: -6
AnSwEr0004: 2
AnSwEr0005: 7
AnSwEr0006: 2
AnSwEr0007: -4
AnSwEr0008: 0
AnSwEr0009: 3
AnSwEr0010: 8
AnSwEr0011: -5
AnSwEr0012: 0
AnSwEr0013: 9
AnSwEr0014: 28
******************************************************
OK, first of all, the webwork uses the '÷' sign for what the book calls 'div'. The mathematical theorem that is the background for all of this is that given any integer a and natural number b, there is an integer d and a natural number r such that 0≤r<b such that
a=d*b+r
Note the important fact that r is strictly non-negative, so that d*b has to be smaller than a. When a is negative, this means that d*b is more negative. For instance, in problems 3 and 4, which were -26÷4 and -26 mod 4, this means that d*4 has to be the largest multiple of 4 less than -26, so that d=-7 not -6, while r has to be 2.
Class tomorrow is cancelled. The exam is rescheduled to next Tuesday.
Hi All,
Class tomorrow Nov. 5 is cancelled. The exam, originally scheduled for Thursday Nov. 7 is rescheduled for Tuesday Nov. 12. Our in-class review session is rescheduled for Thursday Nov. 7--bring your questions and I will answer them.
Class tomorrow Nov. 5 is cancelled. The exam, originally scheduled for Thursday Nov. 7 is rescheduled for Tuesday Nov. 12. Our in-class review session is rescheduled for Thursday Nov. 7--bring your questions and I will answer them.
Sunday, November 3, 2013
Office Hours and Class on Tuesday, Review Session and Exam
1) I'm scheduled to be a workshop all day Tuesday. Therefore:
a) I'll have substitute running the in-class review session. Bring your questions. Better yet, post them here in advance so your fellow students can see what people are wondering about.
b) My office hours on Tuesday are cancelled. Instead I'll have office hours 11am-Noon tomorrow.
2) Apparently there will also be a review session at the math center on 11/5/2013 at 7:00pm--this is all the information I have on this.
3) PS: exam 2 will take place on Thursday.
a) I'll have substitute running the in-class review session. Bring your questions. Better yet, post them here in advance so your fellow students can see what people are wondering about.
b) My office hours on Tuesday are cancelled. Instead I'll have office hours 11am-Noon tomorrow.
2) Apparently there will also be a review session at the math center on 11/5/2013 at 7:00pm--this is all the information I have on this.
3) PS: exam 2 will take place on Thursday.
Thursday, October 31, 2013
homework for next week 11/4/2013-11/8/2013
1) The new webwork sections open this evening and are due next Tuesday at 11:58pm
2) The new book homework is section 6.1, due at the beginning of class on Thursday.
2) The new book homework is section 6.1, due at the beginning of class on Thursday.
Tuesday, October 29, 2013
Thursday, October 24, 2013
homework due 10/31/2013
1) Webwork for sections 2.1-2.4 is now open and is due Thursday 10/31/2013 at midnight.
2) The book homework for sections 5.1 and 5.3 is due Thursday 10/31/2013, AT THE BEGINNING OF CLASS!
2) The book homework for sections 5.1 and 5.3 is due Thursday 10/31/2013, AT THE BEGINNING OF CLASS!
Tuesday, October 22, 2013
The well ordering property
(from Wikipedia)
In mathematics, a well-order relation (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the well-order relation is then called a well-ordered set. The hyphen is frequently omitted in contemporary papers, yielding the spellings wellorder, wellordered, and wellordering.
In mathematics, a well-order relation (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the well-order relation is then called a well-ordered set. The hyphen is frequently omitted in contemporary papers, yielding the spellings wellorder, wellordered, and wellordering.
Wednesday, October 9, 2013
Your grades up to now
Posting ID
|
OffHr
|
HW1
|
HW2
|
HW3
|
HW4
|
HW5
|
ex1
|
%
|
0220-065
|
10
|
10
|
10
|
10
|
10
|
32
|
0.55
|
|
0279-172
|
8
|
10
|
10
|
10
|
10
|
65
|
0.75
|
|
0840-064
|
10
|
8
|
10
|
59
|
0.58
|
|||
0863-521
|
10
|
7
|
10
|
9
|
10
|
35
|
0.54
|
|
1032-842
|
10
|
9
|
10
|
10
|
10
|
10
|
72
|
0.87
|
1104-203
|
10
|
10
|
33
|
0.35
|
||||
1249-800
|
9
|
10
|
9
|
6
|
10
|
0.29
|
||
1305-540
|
10
|
0.07
|
||||||
1396-474
|
10
|
10
|
10
|
10
|
10
|
70
|
0.8
|
|
1410-690
|
10
|
8
|
10
|
8
|
32
|
0.45
|
||
1414-323
|
10
|
6
|
3
|
63
|
0.55
|
|||
1769-125
|
10
|
9.5
|
10
|
10
|
10
|
67
|
0.78
|
|
1937-088
|
10
|
10
|
10
|
53
|
0.55
|
|||
2087-431
|
10
|
10
|
10
|
9
|
6
|
10
|
50
|
0.7
|
2191-657
|
10
|
10
|
10
|
10
|
10
|
10
|
94
|
1.03
|
2460-501
|
10
|
10
|
10
|
10
|
10
|
10
|
91
|
1.01
|
2661-835
|
7.5
|
10
|
10
|
10
|
10
|
83
|
0.87
|
|
2729-431
|
10
|
10
|
9.5
|
10
|
10
|
10
|
61
|
0.8
|
2876-533
|
10
|
10
|
10
|
10
|
10
|
10
|
87
|
0.98
|
3138-233
|
10
|
7.5
|
10
|
10
|
10
|
10
|
68
|
0.84
|
3241-408
|
10
|
9
|
10
|
6
|
10
|
3
|
28
|
0.51
|
3852-035
|
10
|
10
|
6
|
6
|
10
|
72
|
0.76
|
|
3858-648
|
10
|
9.5
|
10
|
10
|
10
|
10
|
40
|
0.66
|
3898-660
|
10
|
10
|
10
|
10
|
10
|
10
|
53
|
0.75
|
4098-316
|
10
|
7.5
|
10
|
6
|
10
|
55
|
0.66
|
|
4296-867
|
10
|
10
|
0.13
|
|||||
4460-459
|
10
|
10
|
10
|
10
|
10
|
58
|
0.72
|
|
4470-016
|
10
|
18
|
0.19
|
|||||
4683-269
|
10
|
10
|
10
|
10
|
10
|
51
|
0.67
|
|
5272-592
|
10
|
7
|
10
|
10
|
9
|
44
|
0.6
|
|
5530-423
|
10
|
6.5
|
3
|
78
|
0.65
|
|||
5695-478
|
10
|
10
|
10
|
10
|
9
|
89
|
0.92
|
|
5701-387
|
10
|
5.5
|
10
|
8
|
8
|
10
|
48
|
0.66
|
5800-194
|
10
|
10
|
49
|
0.46
|
||||
5969-324
|
10
|
10
|
10
|
7
|
80
|
0.78
|
||
6090-307
|
10
|
9.5
|
10
|
10
|
8
|
10
|
77
|
0.9
|
6159-095
|
10
|
10
|
10
|
10
|
10
|
10
|
85
|
0.97
|
6952-749
|
10
|
8
|
10
|
10
|
10
|
70
|
0.79
|
|
7784-916
|
10
|
10
|
10
|
10
|
10
|
10
|
64
|
0.83
|
7840-307
|
10
|
6
|
54
|
0.47
|
||||
8274-004
|
10
|
10
|
10
|
7
|
43
|
0.53
|
||
8478-148
|
10
|
8
|
10
|
0.19
|
||||
8796-274
|
10
|
10
|
10
|
10
|
10
|
10
|
41
|
0.67
|
8867-709
|
10
|
9
|
10
|
9
|
10
|
56
|
0.69
|
|
8893-711
|
9
|
9.5
|
10
|
10
|
10
|
65
|
0.76
|
|
9261-936
|
10
|
31
|
0.27
|
|||||
9356-838
|
10
|
10
|
10
|
10
|
10
|
10
|
71
|
0.87
|
9603-876
|
10
|
10
|
79
|
0.66
|
||||
9788-216
|
10
|
8
|
10
|
10
|
6
|
10
|
94
|
0.99
|
9890-353
|
10
|
7.5
|
10
|
8
|
10
|
10
|
74
|
0.86
|
Tuesday, October 8, 2013
Monday, September 30, 2013
Saturday, September 21, 2013
Homework for next week (9/23-9/28/2013)
1) The webwork for next week (section 1.6) is now open and is due on Tuesday at midnight.
2) the written homework for next week is sections 2.1, 2.2 and 2.3 and is due Thursday at the beginning of class.
By the way, I want to remind you that the first exam is on October 1, which is Tuesday of the following week.
2) the written homework for next week is sections 2.1, 2.2 and 2.3 and is due Thursday at the beginning of class.
By the way, I want to remind you that the first exam is on October 1, which is Tuesday of the following week.
Thursday, September 19, 2013
webwork questions
Hi Dr. Tom! I have some questions about the webwork assignment:
Section1.4 Rosen7: Problem 2
The notation ∃!xP(x) denotes the proposition " There exists a unique x such that P(x) is true. "
If the universe of discourse is the set of integers, what are the truth values of the following?
∃!x(x>1): my answer is true, but the correct answer is false?
<the critical issue here is that there should exist one and only one solution of the inequality. If there is no solution, or if there are more than one solutions, then the answer is false. So, over the integers, how many solutions are there?>
∃!x(x^2=1): my answer is true( 1^2=1?), the correct answer is false?
<Again, the issue is how many solutions are there to the equation x^2=1 over the integers?>
Section1.5 Rosen7: Problem 1
Determine the truth value of the following statements if the universe of discourse of each variable is the set of real numbers:
∃x∀y≠0(xy=1): my answer is true, but the correct answer is false?
<The critical issue here is understanding what the order of quantifiers tells us: ∃x∀y≠0 says that there is some x such that for all y the predicate function xy=1 is true--you have to choose the x and then show that the equation is true for all y≠0. On the other hand ∀y≠0∃x would say that for every y you could find some x--which is a less restrictive statement>
Saturday, September 14, 2013
The Homework for Next Week (9/15/2013-9/21/2013)
1) The written homework for next week is sections 1.6 and 1.7 (see the syllabus)
2) The webwork for next week is open; do sets 1.4 and 1.5; since I didn't open the webwork until just now, I'm giving until Thursday to do it.
2) The webwork for next week is open; do sets 1.4 and 1.5; since I didn't open the webwork until just now, I'm giving until Thursday to do it.
Thursday, September 5, 2013
Homework Due Next Week (September 8-13)
1) The next webwork assignment is now open and is due Tuesday 9/10/2013 at midnight. Remember to get it done correctly by logging in anonymously as "Student" first
2) Written homework sections 1.4 and 1.5 are due at the beginning of class on Thursday 9/12/2013
Monday, September 2, 2013
Webwork Question (bitwise operations)
Hi,
I'm trying to do this problem and I'm not even sure what they're asking us
to do. Can you give me a little guidance please? Thank you.
*********************************************************
This is the problem webwork tells me you are working, is this correct?
Evaluate each of the following expressions:
(a)11000∧(01011∨11011)
(b)(01111∧10101)∨01000
(c)(01010⊕11011)⊕01000
(d)(11011∨01010)∧(10001∨11011)
This problem is about bitwise logical operations, which were covered in section 1.1--You should
reread that section. The general idea is that you consider "1" to be an equivalent of "True", "0" to
be an equivalent of "False" and then perform logical operations "And", "Or" and "XOr" on a
pointwise basis. This link takes you to the wiki article on the subject, and gives more detailed
explanation than the book.
I'm trying to do this problem and I'm not even sure what they're asking us
to do. Can you give me a little guidance please? Thank you.
*********************************************************
This is the problem webwork tells me you are working, is this correct?
Evaluate each of the following expressions:
(a)
(b)
(c)
(d)
reread that section. The general idea is that you consider "1" to be an equivalent of "True", "0" to
Thursday, August 29, 2013
Next Week's Homework Assignment
Written homework: sections 1.2 and 1.3
Webwork section 1.3 (opens on Thursday after class)
Webwork section 1.3 (opens on Thursday after class)
webwork question
"I didn't realize I only had 3 tries on a Web Work problem and couldn't finish it, What should I do?"
This problem can be avoided by first anonymously logging in to webwork as "student", working the problems until you get them correct, and only then logging in as yourself to work the problem correctly so that the your score can be recorded.
Wednesday, August 28, 2013
FIrst Webwork Assignment
The first webwork assignment (section 1.1) will open tomorrow right after class, and close at 11:59PM next Tuesday. Webwork can be accessed at here. The introduction to webwork is also open, and suggested for those who haven't worked with webwork before.
Monday, August 26, 2013
A Puzzle
A census taker approaches a woman leaning on her gate and asks about the ages of her children. She says, "I have three children and the product of their ages is seventy–two. The sum of their ages is the number on this gate." The census taker does some calculation and claims not to have enough information. The woman enters her house, but before slamming the door tells the census taker, "I have to see to my eldest child who is in bed with measles." The census taker departs, satisfied.
What are the ages of the three children?
Sunday, August 25, 2013
Office Hours
Some 38 of 53 MAT243 students responded to the office hours poll and received 10 extra credit points. There were 15 students who did not respond and who did not receive 10 extra credit points....
I just announced the new office hours on the course syllabus by email. They are
Tuesdays Noon-1:00PM
Wednesdays 9:00AM-10:00AM
Fridays 11:00AM-Noon
There were no emails returned as undeliverable, so it looks like electronic communications are all operational. Of course, this doesn't mean that everybody is listening...but I keep hoping for the best outcome.
Some 38 of 53 MAT243 students responded to the office hours poll and received 10 extra credit points. There were 15 students who did not respond and who did not receive 10 extra credit points....
I just announced the new office hours on the course syllabus by email. They are
Tuesdays Noon-1:00PM
Wednesdays 9:00AM-10:00AM
Fridays 11:00AM-Noon
There were no emails returned as undeliverable, so it looks like electronic communications are all operational. Of course, this doesn't mean that everybody is listening...but I keep hoping for the best outcome.
Thursday, August 22, 2013
Hey Dr. Taylor, I got onto Khan Academy to look around and see what was on there but I don't see anything for discrete math. Am I just looking in the wrong spot or is it under another name?
Try searching on specific terms that were used in the lecture and/or the textbook. For example, here's a movie and some exercises directly related to the lecture today.
Conditional statements and deductive reasoning
Conditional statements exercise examples
Conditional statements
Logical argument and deductive reasoning exercise example
Logical arguments and deductive reasoning
Conditional statements and truth value
Logical reasoning
Hi all,
Welcome to my MAT243 blog for Fall 2013. There are more than 50 students in this course. This means that we will have to keep a fairly rigid schedule with the lectures (normally I like to slow down or speed up according to feedback from the students).
This week and next week we'll be concerned about propositional logic--more about this in class. This is the classical logic in which propositions take two values only--true and false--and in which the truth values of compound propositions are determined from precise rules involving the truth values of the components. Note that a binary valued variable also takes two values, zero and one. There is an important and useful connection between logic and binary arithmetic according to the rule True->1, False->0. This means that, although a major purpose of propositional logic is learning how to think, it also serves as the backdrop for many calculations involving binary numbers, that is with the inner workings of computers.
Welcome to my MAT243 blog for Fall 2013. There are more than 50 students in this course. This means that we will have to keep a fairly rigid schedule with the lectures (normally I like to slow down or speed up according to feedback from the students).
This week and next week we'll be concerned about propositional logic--more about this in class. This is the classical logic in which propositions take two values only--true and false--and in which the truth values of compound propositions are determined from precise rules involving the truth values of the components. Note that a binary valued variable also takes two values, zero and one. There is an important and useful connection between logic and binary arithmetic according to the rule True->1, False->0. This means that, although a major purpose of propositional logic is learning how to think, it also serves as the backdrop for many calculations involving binary numbers, that is with the inner workings of computers.
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