Math Think

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Hi there. I'm doing Stanford's Introduction to Mathematical Thinking. Following are my notes and assignment answers.

Resources

The course textbook is Introduction to Mathematical Thinking by Keith Devlin.

Week 1

Assignment 1

Question 1

Q: 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.

A:

  • The man with the telescope saw the woman
  • The woman with the telescope was seen by the man

Question 2

Q: For each of the three ambiguous newspaper headlines I stated in the lecture, rewrite it in a way that avoids the amusing second meaning, while retaining the brevity of a typical headline:

  1. Sisters reunited after ten years in checkout line at Safeway.
  2. Large hole appears in High Street. City authorities are looking into it.
  3. Mayor says bus passengers should be belted.

A:

  1. After ten years apart, Sisters reunited in checkout line at Safeway.
  2. Large hole appears in High Street. City authorities are investigating.
  3. Mayor says bus passengers should wear seatbelts.

Question 3

Q: The following notice was posted on the wall of a hospital emergency room:

No head injury is too trivial to ignore.

Reformulate to avoid the unintended second reading. (The context for this sentence is so strong that many people have difficulty seeing there is an alternative meaning.)

A:

All head injuries are serious and require investigation.

Question 4

Q: You often see the following notice posted in elevators:

In case of fire, do not use elevator.

This one always amuses me. Comment on the two meanings and reformulate to avoid the unintended second reading. (Again, given the context for this notice, the ambiguity is not problematic.)

A: The notice is not suggesting that the elevator can be used to combat the fire.

If there is a fire do not travel using the elevator.

Question 5

Official documents often contain one or more pages that are empty apart from one sentence at the bottom:

This page intentionally left blank.

Does the sentence make a true statement? What is the purpose of making such a statement? What reformulation of the sentence would avoid any logical problems about truth? (Once again, the context means that in practice everyone understands the intended meaning and there is no problem. But the formulation of a similar sentence in mathematics at the start of the twentieth century destroyed one prominent mathematician’s seminal work and led to a major revolution in an entire branch of mathematics.)

Q: Does the sentence make a true statement?

A: No.

Q: What is the purpose of making such a statement?

A: That the otherwise blank page is there deliberately.

Q: What reformulation of the sentence would avoid any logical problems about truth?

A: This page intentionally left blank, sans this notice.

Question 6

Q: Find (and provide citations for) three examples of published sentences whose literal meaning is (clearly) not what the writer intended. (This is much easier than you might think. Ambiguity is very common.)

A: My answers taken from the book titles listed on the Wikipedia page of Professor Devlin:

  1. Goodbye, Descartes: the End of Logic and the Search for a New Cosmology of the Mind
    • What has ended? Logic? Or Logic and the Search for a New Cosmology of the Mind?
  2. Life by the Numbers
    • Life beside the numbers? Life per the numbers?
  3. The Man of Numbers: Fibonacci's Arithmetic Revolution
    • Fibonacci rotated arithmetic 360 degrees? 6.2832 radians? What does that even mean?

Question 7

Q: Comment on the sentence "The temperature is hot today." You hear people say things like this all the time, and everyone understands what is meant. But using language in this sloppy way in mathematics would be disastrous.

A: It's the environment on the day which is hot; the temperature is a measure of the heat, not a substance which can be heated.

Question 8

Q: How would you show that not every number of the form N = (p_1 \cdot p_2 \cdot p_3 \cdot ... \cdot p_n) + 1 is prime, where p_1, p_2, p_3, ..., p_n, ... is the list of all prime numbers?

A: All that is required is one counter example. So I wrote a computer program to find one... the program is in JavaScript, I ran it with Node.js. The program is not algorithmically efficient, but it was programming-time efficient, and I do believe it is correct.

 function main() {

   let list = [], n = 0;

   for ( let p of list_primes( 100 ) ) {

     n++;

     console.log( 'p_' + n + ' = ' + p );

     list.push( p );

     let test = multiply( list ) + 1;

     if ( is_prime( test ) ) { continue; }

     return console.log( 'test ' + test + ' is not prime for p_n = ' + p );

   }
 }

 function multiply( list ) {

   let result = 1;

   for ( let n of list ) {

     result *= n;

   }

   return result;

 }

 function* list_primes( max ) {

   let n = 1;

   while ( ++n <= max ) {

     if ( is_prime( n ) ) { yield n; }

   }
 }

 function is_prime( n ) {

   for ( let i = 2; i < n; i++ ) {

     if ( n % i === 0 ) { return false; }

   }

   return true;

 }

 main();

The output is:

p_1 = 2
p_2 = 3
p_3 = 5
p_4 = 7
p_5 = 11
p_6 = 13
test 30031 is not prime for p_n = 13

Just for fun 1

Q: Provide a context and a sentence within that context, where the word and occurs five times in succession, with no other word between those five occurrences. (You are allowed to use punctuation.)

A:

Context: find logical connectives A, B, and C where: T A T B T = T C T

Sentence: the solution could be determined using "'and' and 'and'" and "and"...

Just for fun 2

Q: Provide a context and a sentence within that context, where the words and, or, and, or, and occur in that order, with no other word between them. (Again, you can use punctuation.)

Reply to Jevan

In response to Week 1 Assignment 1, Jevan Pipitone's Answers:

Hi Jevan! Good to see you here. Thought I might compare your answers to mine, which are over here: https://www.jj5.net/sixsigma/Math_Think

1. we're in agreement

2. our answers are very similar

3. oh, I think you might be right there. I took the meaning as "an injury where a person has been decapitated is too trivial to ignore", which is not a trivial injury at all! I wonder what was intended there?

4. we're in disagreement again! You were considering a "fire in a suitcase" whereas I thought the other meaning was "if there is a fire don't try to put it out with the elevator"

5. I didn't assume the blank page was necessarily for note taking, just perhaps for best formatting. The purpose of notifying the reader of the intentional blank page was so they didn't worry that they were missing something.

6. where you investigated misapplications of the term 'literally' I thought it would be fun to try and find potential ambiguity in the title of some of Prof. Devlin's books: https://www.jj5.net/sixsigma/Math_Think#Question_6

7. I'm not sure the issue is the lack of specificity of a temperature reading, but the fact that "temperature" is a measure, not a substance which can be heated. It's not the temperature which is hot, but the environment on the given day.

8. we're in total agreement here. But I was lazy and wrote a program to do the calculation for me https://www.jj5.net/sixsigma/Math_Think#Question_8

JFF 1. I think you need to re-read the requirements there Jevan, they ask for the five ands to be separated only by punctuation. I had a go at this but I'm not really happy with my results: https://www.jj5.net/sixsigma/Math_Think#Just_for_fun_1

JFF 2. I haven't come up with an answer for this yet. I'll try to give it some brain-space tomorrow.

Assignment 2

Question 1

Simplify the following symbolic statements as much as you can, leaving your answer in the standard symbolic form.

# Question Answer
a) (\pi > 0) \and (\pi < 10) 0 < \pi < 10
b) (p \geq 7) \and (p < 12) 7 \leq p < 12
c) (x < 5) \and (x < 7) x < 7
d) (x < 4) \and (x < 6) x < 6
e) (y < 4) \and (y^2 < 9) y^2 < 9
f) (x \geq 0) \and (x \leq 0) x = 0

Question 2

Express each of your simplified statements from question 1 in natural English.

# Expression English
a) 0 < \pi < 10 π is greater than zero and less than ten.
b) 7 \leq p < 12 p is greater than or equal to seven and less than twelve.
c) x < 7 x is less than seven.
d) x < 6 x is less than six.
e) y^2 < 9 y squared is less than nine.
f) x = 0 x equals zero.

Question 3

Q: What strategy would you adopt to show that the conjunction \phi_1 \and \phi_2 \and ... \and \phi_n is true?

A: The conjunction is true if all \phi_n are true.

Question 4

Q: What strategy would you adopt to show that the conjunction \phi_1 \and \phi_2 \and ... \and \phi_n is false?

A: The conjunction is false if any \phi_n is false.

Question 5

Simplify the following symbolic statements as much as you can, leaving your answer in a standard symbolic form:

# Question Answer
a) (\pi > 3) \or (\pi > 10) \pi > 3
b) (x < 0) \or (x > 0) x \neq 0
c) (x = 0) \or (x > 0) x \geq 0
d) (x > 0) \or (x \geq 0) x \geq 0
e) (x > 3) \or (x^2 > 9) x > 3

Question 6

Express each of your simplified statements from question 5 in natural English.

# Expression English
a) \pi > 3 π is greater than three.
b) x \neq 0 x is not zero.
c) x \geq 0 x is greater than or equal to zero.
d) x \geq 0 x is greater than or equal to zero.
e) x > 3 x is greater than three.

Question 7

Q: What strategy would you adopt to show that the disjunction \phi_1 \or \phi_2 \or ... \or \phi_n is true?

A: The disjunction is true if any \phi_n is true.

Question 8

Q: What strategy would you adopt to show that the disjunction \phi_1 \or \phi_2 \or ... \or \phi_n is false?

A: The disjunction is false if all \phi_n are false.

Question 9

Simplify the following symbolic statements as much as you can, leaving your answer in a standard symbolic form:

# Question Answer
a) \neg(\pi > 3.2) \pi <= 3.2
b) \neg(x < 0) x \geq 0
c) \neg(x^2 > 0) x = 0
d) \neg(x = 1) x \neq 1
e) \neg\neg\psi \psi

Question 10

Express each of your simplified statements from question 9 in natural English.

# Expression English
a) \pi <= 3.2 π is less than or equal to three point two.
b) x \geq 0 x is greater than or equal to zero.
c) x = 0 x is zero.
d) x \neq 1 x is not equal to one.
e) \psi ψ is true.

Question 11

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?

Letter Statement
D The dollar is strong
Y The Yuan is strong
T New US–China trade agreement signed
# English Logic
a) Dollar and Yuan both strong D \and Y
b) Yuan weak despite new trade agreement, but Dollar remains strong \neg{Y} \and T \and D
c) Dollar and Yuan can’t both be strong at same time \neg{(D \and Y)}
d) New trade agreement does not prevent fall in Dollar and Yuan T \and \neg{D} \and \neg{Y}
e) US–China trade agreement fails but both currencies remain strong \neg{T} \and D \and Y

Assignment 2 thinking 1

Q: 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 affairs 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?

A: I think that 'not' and '¬' have the same meaning. I think what might differ are our definitions for "guilty" and "proven", whether we mean them literally, or in a technical sense.

Assignment 2 thinking 2

Q: The truth table for ¬¬φ is 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?

A: To keep things formal we'd probably need to introduce some quantification for our degree pleasure, as, e.g., a number from 1 to 10, a number of stars, etc.