I didn't get the quiz completed yesterday. Although I had done most of the work (on paper) over the weekend, I just didn't have time Sunday night to finish it. Yesterday, I had the "privelege" of sitting in a courtroom for three hours because I am being sued in small claims by someone that I should be suing. They never showed up.
What do I do? Do I keep on working on this class? Do I drop it? My life was terribly out-of-control last week. I belong to two Toastmasters clubs. On Wednesday, I teach basic math and computer skills to a group at my church. (Because our church area ["ward" to those who know how the LDS church works] encompasses the poorest parts of Atlanta, many of our members are low-income and low-educated. A group of educated people in the ward have been assigned to help stop the cycle of poverty that these otherwise wonderful people have experienced.) On Thursday, my son was in town for a few hours. The advantage of living near the world's busiest airport -- you get unexpected visits from friends and family when they have layovers.
Anyone out there have feelings about this? Is this class more work than YOU expected? I am also bogged down by the nomenclature -- I have always thought in terms of "unions" and "and" and "or", not phis and psis. Plus, the use of pi as a variable really threw me the first week. Pi is 3.14159..., not some random variable.
My rant for the day.
Tuesday, September 17, 2013
Saturday, September 14, 2013
Assignment 4: Questions 12, 13, and 14.
12. Write down the contrapositives:
(a) If two rectangles are congruent, they have the same area.
(b) If a triangle with sides a, b, c, is right-angles, then a^2 + b^2 = c^2
(c) If 2^n -1 is prime, then n is prime.
(d) If the Yuan rises, the Dollar will fall.
13. Use truth tables to show that the contrapositive and the converse of A => B are not equivalent.
14. Write down the converses of the four statements in question 12.
(a) If two rectangles are congruent, they have the same area.
(b) If a triangle with sides a, b, c, is right-angles, then a^2 + b^2 = c^2
(c) If 2^n -1 is prime, then n is prime.
(d) If the Yuan rises, the Dollar will fall.
13. Use truth tables to show that the contrapositive and the converse of A => B are not equivalent.
14. Write down the converses of the four statements in question 12.
Assignment 4: Question 11
Use truth tables to prove the equivalence of A => B and (not B) => (not A)
Assignment 4: Questions 7, 8, 9, and 10.
7. Show that A <=> B is equivalent to (not A) <=> (not B).
8. Construct truth tables to illustrate the following:
(a) A <=> B.
(b) A => (B and C).
9. Use truth tables to prove that the following are equivalent: A => (B and C) and (A => B) and (A => C).
10. Verify the equivalence in question 9 by means of a logical argument.
8. Construct truth tables to illustrate the following:
(a) A <=> B.
(b) A => (B and C).
9. Use truth tables to prove that the following are equivalent: A => (B and C) and (A => B) and (A => C).
10. Verify the equivalence in question 9 by means of a logical argument.
Assignment 4: Question 6
Give a natural sounding denial of each of the following statements.
(a) 34,159 is a prime number.
(b) Roses are red and violets are blue.
(c) If there are no hamburgers, I'll have a hot dog.
(d) Fred will go but he will not play.
(e) The number x is either negative or greater than 10.
(f) We will win the first game or the second.
(a) 34,159 is a prime number.
(b) Roses are red and violets are blue.
(c) If there are no hamburgers, I'll have a hot dog.
(d) Fred will go but he will not play.
(e) The number x is either negative or greater than 10.
(f) We will win the first game or the second.
Assignment 4: Question 5
(See assignment).
Question: Provide an analogous logical argument to show that not(A and B) and (not A) or (not B) are equivalent.
Question: Provide an analogous logical argument to show that not(A and B) and (not A) or (not B) are equivalent.
Assignment 4: Question 4
(a) Construct a truth table for the logical statement:
[A and (A => B)] => B
(b) Explain how the truth table to obtain demonstrates that modus ponens is a valid rule of inference.
[A and (A => B)] => B
(b) Explain how the truth table to obtain demonstrates that modus ponens is a valid rule of inference.
Assignment 4: Questions 2 and 3
I will just post these questions now, and post my feeble attempts later.
2. Build a truth table to show that (A => B) <=> (not B or A) is true for all truth values of A and B.
3. Build a truth table to show that (A does not imply B) is equivalent to (A and not B) is a tautology.
2. Build a truth table to show that (A => B) <=> (not B or A) is true for all truth values of A and B.
3. Build a truth table to show that (A does not imply B) is equivalent to (A and not B) is a tautology.
Assignment 4, finally - question 1
Sorry, I got behind. But, none of you seemed to be "yelling" at me, so I hope your life was a little less hectic than mine this week.
Problem 1:
Build a truth table to prove that A <=> B is true if A and B are both true or both false, and A <=> B is false if exactly one of A, B is true and the other false.
My answer:
A | B | A => B | B => A | (A => B) and (B => A)
T | T | T | T | T
T | F | F | T | F
F | T | T | F | F
F | F | T | T |T
Your thoughts?
Problem 1:
Build a truth table to prove that A <=> B is true if A and B are both true or both false, and A <=> B is false if exactly one of A, B is true and the other false.
My answer:
A | B | A => B | B => A | (A => B) and (B => A)
T | T | T | T | T
T | F | F | T | F
F | T | T | F | F
F | F | T | T |T
Your thoughts?
Thursday, September 12, 2013
Assignment 3 - Problems 4 and 5
Complete the truth table:
A | B | not B | A => B | A not implies > B | A and not B
T | T | F | T | F | F
T | F | T | F | T | T
I will leave this problem for now and edit it later on. It looks, though, like (A does not imply B) and (A and not B) are the same.
A | B | not B | A => B | A not implies > B | A and not B
T | T | F | T | F | F
T | F | T | F | T | T
I will leave this problem for now and edit it later on. It looks, though, like (A does not imply B) and (A and not B) are the same.
Tuesday, September 10, 2013
Assignment 3 - Problems 2 and 3
I am tired of phi's and pi's and psi's, so am going to use A and B
My answers for the truth table
A | not A | B | A implies B | Not A and B
T | F | T | T | F
T | F | F | F | F
F | T | T | T | T
F | T | F | T | F
I guess I need to go to bed. Did I mess up somewhere? Let me know.
My answers for the truth table
A | not A | B | A implies B | Not A and B
T | F | T | T | F
T | F | F | F | F
F | T | T | T | T
F | T | F | T | F
I guess I need to go to bed. Did I mess up somewhere? Let me know.
Assignment 3 - Problem 1
D = The dollar is strong
Y = The Yuan is strong
T = Trade Agreement signed
Here is my answer for the first part of #1:
(a) T => D ^ Y
(b) D => ~ Y
(c) ~D => ~T
(d) T => ~D V ~Y
Someone else's turn....
Y = The Yuan is strong
T = Trade Agreement signed
Here is my answer for the first part of #1:
(a) T => D ^ Y
(b) D => ~ Y
(c) ~D => ~T
(d) T => ~D V ~Y
Someone else's turn....
Lecture 3
This was a tough lecture to listen to, but here is the bottom line:
If one truth implies another truth, the implication is true.
If the truth implies a falsehood, the implication must be false. A single truth cannot imply both a truth and a falsehood.
If a falsehood implies either a truth or a falsehood, the implication must be true. The antecedent must be true to imply anything.
I will post assignment 3 here in a few minutes. Ugly story: My house was broken into this summer and all of my electronics were stolen. I replaced my nice laptop with a new one that has Windows 8, a great operating system for touchscreens but it sucks for laptops. Because Adobe Acrobat is considered a "Windows App", Windows 8 only allows it to be shown full screen. I can't look at the PDF, then copy and paste the text into this blog, or even type as I am looking at it. Because my printer is old (it takes a while to replace electronics!), it won't talk to Windows 8. So, I need to email the PDF to my son (whose computer WILL talk to the printer), get him to print the PDF, and then type.
Ugh!
If one truth implies another truth, the implication is true.
If the truth implies a falsehood, the implication must be false. A single truth cannot imply both a truth and a falsehood.
If a falsehood implies either a truth or a falsehood, the implication must be true. The antecedent must be true to imply anything.
I will post assignment 3 here in a few minutes. Ugly story: My house was broken into this summer and all of my electronics were stolen. I replaced my nice laptop with a new one that has Windows 8, a great operating system for touchscreens but it sucks for laptops. Because Adobe Acrobat is considered a "Windows App", Windows 8 only allows it to be shown full screen. I can't look at the PDF, then copy and paste the text into this blog, or even type as I am looking at it. Because my printer is old (it takes a while to replace electronics!), it won't talk to Windows 8. So, I need to email the PDF to my son (whose computer WILL talk to the printer), get him to print the PDF, and then type.
Ugh!
Monday, September 9, 2013
This weeks quiz
Anybody else try this week's quiz yet? https://class.coursera.org/maththink-003/quiz/index?quiz_type=homework
I did it last night. I'm not sure how I did. I have always been much more comfortable giving tests than taking them. Thoughts on the test? (No, I am not asking for what you answered, just your thoughts).
I did it last night. I'm not sure how I did. I have always been much more comfortable giving tests than taking them. Thoughts on the test? (No, I am not asking for what you answered, just your thoughts).
Thursday, September 5, 2013
A little about myself
I am an "older" student living in Atlanta, GA. I am originally from Seattle, WA (USA) and have been in Atlanta for two years.
I work in the IT industry as a business analyst. My passion is requirements -- if you want a new computer application, web site, etc., what do you want it to do? I find out from you and then explain your needs to the programmers. After they are done, I make sure they built what you needed. An over-simplification of what I do. Mathematical thinking helps me immensely in my career.
I am a former college algebra teacher, but that doesn't make me an expert. I am learning just as much as any of you.
I have two grown sons and many others that have been mine for short times (in other words, I didn't give birth to them, but they call me Mom.) I have 3 granddaughters and a grandson ranging in age from 3 months to nearly 6 years old.
Hope to get some lively discussions going here!
I work in the IT industry as a business analyst. My passion is requirements -- if you want a new computer application, web site, etc., what do you want it to do? I find out from you and then explain your needs to the programmers. After they are done, I make sure they built what you needed. An over-simplification of what I do. Mathematical thinking helps me immensely in my career.
I am a former college algebra teacher, but that doesn't make me an expert. I am learning just as much as any of you.
I have two grown sons and many others that have been mine for short times (in other words, I didn't give birth to them, but they call me Mom.) I have 3 granddaughters and a grandson ranging in age from 3 months to nearly 6 years old.
Hope to get some lively discussions going here!
Wednesday, September 4, 2013
Assignment 1 -- Question 6
Just saw this on msnbc.com:
http://news.msn.com/world/1-woman-dies-every-hour-over-dowry-in-india
Not only is she dying every hour, but she is falling on her dowry each time!
My sons' high school had a sign that always puzzled me as well: "No Parking Cars
Will Be Impounded." I always thought it would be very safe to park there.
When I was growing up, Seattle had several signs that said, "High pedestrian collision location." The sign was clear, but I loved the picture of a figure bouncing off the hood of a car. http://news.google.com/newspapers?nid=1499&dat=19800323&id=lGMaAAAAIBAJ&sjid=6SoEAAAAIBAJ&pg=4453,2001989
http://news.msn.com/world/1-woman-dies-every-hour-over-dowry-in-india
Not only is she dying every hour, but she is falling on her dowry each time!
My sons' high school had a sign that always puzzled me as well: "No Parking Cars
Will Be Impounded." I always thought it would be very safe to park there.
When I was growing up, Seattle had several signs that said, "High pedestrian collision location." The sign was clear, but I loved the picture of a figure bouncing off the hood of a car. http://news.google.com/newspapers?nid=1499&dat=19800323&id=lGMaAAAAIBAJ&sjid=6SoEAAAAIBAJ&pg=4453,2001989
Assignment 2 - Questions to think about
In US law, a trial verdict of "Not guilty" is given when the prosecution fails to prove guilt. This, of
course, does not mean the defendant is, as a matter of actual fact, innocent. Is this state of a airs
captured accurately when we use "not" in the mathematical sense? (i.e., Do "Not guilty" and ~guilty" mean the same?) What if we change the question to ask if "Not proven" and ~ proven"
mean the same?
=======================================================================
course, does not mean the defendant is, as a matter of actual fact, innocent. Is this state of a airs
captured accurately when we use "not" in the mathematical sense? (i.e., Do "Not guilty" and ~guilty" mean the same?) What if we change the question to ask if "Not proven" and ~ proven"
mean the same?
=======================================================================
The truth table for ~ ~ Φis clearly the same as that for Φ itself,
so the two expressions make identical
truth assertions. This is
not necessarily true for negation in everyday life. For example, you might
find yourself saying “I
was not displeased with the movie." In terms of formal negation, this has
the form ~(~ pleased), but
your statement clearly does not mean that you were pleased with the
movie. Indeed, it means
something considerably less positive. How would you capture this kind of
use of language in the formal framework we
have been looking at?
Assignment 2 - Question 11
Let D be the statement "The dollar is strong", Y the statement "The Yuan is strong" and T the statement "New US-China trade agreement signed". Express the main content of each of the following (fictitious) newspaper headlines in logical notation. (Note that logical notation captures truth, but not the many nuances and inferences of natural language.) How would you justify and defend your answers?
Assignment 2 - Questions 9 and 10
Simplify the following symbolic statements as much as you can, leaving your answer in a standard symbolic form. Express each of your answers in natural English.
Assignment 2 - Questions 7 and 8
What strategy would you adopt to show that the disjunction is true? False?
Assignment 2 - questions 5 and 6
Simplify the following symbolic statements, leaving your answer in the standard symbolic form. Then, express your simplified statement in natural English.
I haven't done this one yet. You get to go first! :)
I haven't done this one yet. You get to go first! :)
Assignment 2 - Questions 3 and 4
What strategy would you adopt to show that the conjunction is true? False?
Assignment 2 - Questions 1 and 2
Simplify the following symbolic statements, leaving your answer in the standard symbolic form. Then, express your simplified statement in natural English.
I haven't done this one yet. You get to go first! :)
I haven't done this one yet. You get to go first! :)
Assignment 1 - Question 8
How would you show that not every number of the form N =(p1*p2*p3*...*pn)+1 is prime?
====================================================================
This theorem says to multiply all of the prime numbers up to a certain level, then add 1. See if the result is also prime. If we can find even one exception, we have disproven the theorem.
Prime: A number is prime if it has exactly two factors -- 1 and itself.
All composite numbers (non-prime numbers) have pairs of factors, where one of the factors is less than the square root of the number, and the second is greater than the square root of the number. The exception is a perfect square, which has a "singular" factor.
For example, 6 has factors of 1 * 6 and 2 * 3; 16 has factors of 1 * 16, 2 * 8, and 4 * 4.
To find if the result from our theorem is prime, multiply the numbers, add 1, then check the square root of the number. Are there any possible primes that we have to look at? We don't need to look at any of the primes that we used to get there, since these numbers can't be factors (if we divided by any of these numbers, we would always get a remainder of 1).
Possible numbers:
2 * 3 + 1 = 7.
2 * 3 * 5 + 1 = 31
2 * 3 * 5 * 7 + 1 = 211
2 * 3 * 5 * 7 * 11 + 1 = 2311
2 * 3 * 5 * 7 * 11 * 13 + 1 = 30031
Are any of these known primes? Here is a list: http://en.wikipedia.org/wiki/List_of_prime_numbers#The_first_500_prime_numbers
The first four results are on this list. How about the last number? The square root of 30031 is 173.29. We need to check every prime less than 173 to find out if it is a factor of 30031. We also know that the primes up to 13 are not factors.
After a little experimentation, you find that 59 * 509 = 30031, so this number is not a prime number.
====================================================================
This theorem says to multiply all of the prime numbers up to a certain level, then add 1. See if the result is also prime. If we can find even one exception, we have disproven the theorem.
Prime: A number is prime if it has exactly two factors -- 1 and itself.
All composite numbers (non-prime numbers) have pairs of factors, where one of the factors is less than the square root of the number, and the second is greater than the square root of the number. The exception is a perfect square, which has a "singular" factor.
For example, 6 has factors of 1 * 6 and 2 * 3; 16 has factors of 1 * 16, 2 * 8, and 4 * 4.
To find if the result from our theorem is prime, multiply the numbers, add 1, then check the square root of the number. Are there any possible primes that we have to look at? We don't need to look at any of the primes that we used to get there, since these numbers can't be factors (if we divided by any of these numbers, we would always get a remainder of 1).
Possible numbers:
2 * 3 + 1 = 7.
2 * 3 * 5 + 1 = 31
2 * 3 * 5 * 7 + 1 = 211
2 * 3 * 5 * 7 * 11 + 1 = 2311
2 * 3 * 5 * 7 * 11 * 13 + 1 = 30031
Are any of these known primes? Here is a list: http://en.wikipedia.org/wiki/List_of_prime_numbers#The_first_500_prime_numbers
The first four results are on this list. How about the last number? The square root of 30031 is 173.29. We need to check every prime less than 173 to find out if it is a factor of 30031. We also know that the primes up to 13 are not factors.
After a little experimentation, you find that 59 * 509 = 30031, so this number is not a prime number.
Assignment 1 - Questions 2 through 5
2. (a) Sisters reunited after 10 years in checkout line at Safeway
(b) Large hole appears in High Street. City authorities are looking into it.
(c) Mayor says bus passengers should be belter.
Better:
(a) After 10 years separation, sisters meet in checkout line
(b) City authorities investigating cause of large hole in High Street
(c) Mayor says bus passengers should wear seatbelts
3. Original: "No head injury is too trivial to ignore."
Better: "Treat all head injuries, no matter how small"
4. "In case of fire, do not use elevator."
This one reminds me of the limited-English-speaker at JFK airport many years ago who wanted a smoke and didn't have a match. He pulled the lever that said, "Pull lever in case of fire."
A whole new meaning appears when you think of defining a "case of fire" and how a person would fit in that case.
5. "This page intentionally left blank." Except for this statement, of course.
Better: "Except for this sentence, this page is intentionally left blank."
(b) Large hole appears in High Street. City authorities are looking into it.
(c) Mayor says bus passengers should be belter.
Better:
(a) After 10 years separation, sisters meet in checkout line
(b) City authorities investigating cause of large hole in High Street
(c) Mayor says bus passengers should wear seatbelts
3. Original: "No head injury is too trivial to ignore."
Better: "Treat all head injuries, no matter how small"
4. "In case of fire, do not use elevator."
This one reminds me of the limited-English-speaker at JFK airport many years ago who wanted a smoke and didn't have a match. He pulled the lever that said, "Pull lever in case of fire."
A whole new meaning appears when you think of defining a "case of fire" and how a person would fit in that case.
5. "This page intentionally left blank." Except for this statement, of course.
Better: "Except for this sentence, this page is intentionally left blank."
Assignment 1 - Question 1
Find two unambiguous (but natural sounding) sentences equivalent to the sentence The man saw the woman with a telescope, the first where the man has the telescope, the second where the woman has the telescope.
=======================================================================================
My answers:
Looking through the telescope, the man saw the woman.
The man saw the woman who was using a telescope.
=======================================================================================
My answers:
Looking through the telescope, the man saw the woman.
The man saw the woman who was using a telescope.
First post
Join me here for an Introduction to Mathmematical Thinking Study Group. I will mention this blog on our class forum. No formal invitatons!
I will post my answers to the first assignment later today or tonight (depending on how busy I get at work :).
Let's just see how this goes.
Diane
I will post my answers to the first assignment later today or tonight (depending on how busy I get at work :).
Let's just see how this goes.
Diane
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