*All course documents available here.*

(a) Not all precious stones are beautiful. (All stones)

(b) All precious stones are not beautiful. (All stones)

(c) Some isosceles triangle is a right triangle. (All triangles)

(d) No right triangle is isosceles. (All triangles)

(e) All people are honest or no one is honest. (All people)

(f) Some people are honest and some people are not honest. (All people)

(g) There is a smallest positive integer. (Real numbers)

(h) Between any real number and any larger real number, there is a rational number. (Real numbers)

(i) No one loves everybody. (All people)

(j) Everybody loves someone. (All people)

(k) For every positive real number x, there is a unique real number y suchthat 2 = x. (Real numbers)

(I) For every complex number, there is at least one complex number such that the product of the two complex numbers is a real number. (Real numbers, as the coefficients)

(m) For every nonzero complex number, there is a unique complex number such that their product is π. (Real numbers, as the coefficients)

1.3.2 For each of the propositions in exercise I, write a useful denial, and give a translation into ordinary English. (Solution not available yet; Solve it!)

1.3.3 Give two proofs of Theorem 1.3(b). (Solution not available yet; Solve it!)

Theorem 1.3(b): If A (x) is an open sentence with variable x, then ¬(∃x)A(x) is equivalent to (∀x)¬A(x).

1.3.4 Which of the following are true? The universe for each is given in parentheses. (Solution not available yet; Solve it!)

(a) (∀x)(x + x ≥ x) (Real numbers)

(b) (∀x)(x + x ≥ x) (Natural numbers)

(c) (∃x)(2x + 3 = 6x + 7) (Natural numbers)

(d) (∃x)(3^x = x^2 ) (Real numbers)

(e) (∃x)(3^x = x) (Real numbers)

(f) (∃x)(3(2 – x) = 5 + 8(1 – x)) (Real numbers)

(g) (∀x)(x^2 + 6x + 5 ≥ 0) (Real numbers)

(h) (∀x)(x^2 + 4x +5 ≥ 0) (Real numbers)

(i) (∃x)(x^2 + x + 41 is prime) (Natural numbers)

(j) (∀x)(x^2 + x + 41 is prime) (Natural numbers)

(k) (∀x)(x^3 + 17x^2 + 6x + 100 ≥ 0) (Real numbers)

1.3.5 Give an English translation for each. The universe is given in parentheses. (Solution not available yet; Solve it!)

(a) (∀x)(x ≥ I) (Natural numbers)

(b) (∃!x)(x ≥ 0 ∧ x ≤ 0) (Real numbers)

(c) (∀x)( x is prime ∧ x ≠ 2 ⇒ x is odd) (Natural numbers)

(d) (∃!x)(ln(x) = 1) (Real numbers)

(e) ¬(∃x)(x^2 < 0 ) (Real numbers)

(f) (∃!x)(x^2 = 0) (Real numbers)

(g) (∀x)(x is odd ⇒ x^2 is odd) (Natural numbers)

1.3.6 Which of the following are true for the universe of all real numbers? (Solution not available yet; Solve it!)

(a) (∀x)(∃y)(x + y = 0).

(b) (∃x)(∀y)(x + y = 0).

(c) (∃x)(∃y)(x^2 + y^2 = -1)

(d) (∀x)[x > 0 ⇒ (∃y)(y < 0 ∧ xy > 0)]

(e) (∀y)(∃x)(∀z)(xy = xz)

(f) (∃x)(∀y)(x ≤ y)

(g) (∀y)(∃x)(x ≤ y)

(h) (∃!y)(y < 0 ∧ y + 3 > 0)

(i) (∃!x)(∀y)(x = y^2)

(j) (∀y)(∃!x)(x = y^2)

(k) (∃!x)(∃!y)(∀w)(w^2 > x – y)

1.3.7 (a) Give a proof of (∃!x)P(x) ⇒ (∃x)P(x). (b) Show that the converse of the conditional sentence in (a) is false. (Solution not available yet; Solve it!)

1.3.8 Write a symbolic translation of the Mean Value Theorem from calculus. (Solution not available yet; Solve it!)

1.3.9 Write a symbolic translation of the definition of limf(x) = L, x->a. Find a useful denial and give an idiomatic English version. (Solution not available yet; Solve it!)

1.3.10 Which of the following are denials of (∃!x)P(x)? (Solution not available yet; Solve it!)

(a) (∀x)(P(x))∨(∀x)(¬P(x))

(b) (∀x)(¬P(x))∨(∃y)(∃z)(y ≠ z ∧ P(y) ∧ P(z))

(c) (∀x)[P(x) ⇒ (∃y)(P(y) ∧ x ≠ y)]

(d) ¬(∀x)(∀y)[(P(x) ∧ P(y)) ⇒ x = y]

1.3.11 Give a denial of “You can fool some of the people all of the time and all of the people some of the time, but you cannot fool all of the people all of the time.” (Solution not available yet; Solve it!)

1.3.12 Riddle: What is the English translation of the symbolic statement ∀∃∃∀? (Solution not available yet; Solve it!)

*All course documents available here.*

1.3.1 Translate the following English sentences into symbolic sentences with quantifiers. The universe for each is given in parentheses.

(a) Not all precious stones are beautiful. (All stones)

(b) All precious stones are not beautiful. (All stones)

(c) Some isosceles triangle is a right triangle. (All triangles)

(d) No right triangle is isosceles. (All triangles)

(e) All people are honest or no one is honest. (All people)

(f) Some people are honest and some people are not honest. (All people)

(g) There is a smallest positive integer. (Real numbers)

(h) Between any real number and any larger real number, there is a rational number. (Real numbers)

(i) No one loves everybody. (All people)

(j) Everybody loves someone. (All people)

(k) For every positive real number x, there is a unique real number y suchthat 2 = x. (Real numbers)

(I) For every complex number, there is at least one complex number such that the product of the two complex numbers is a real number. (Real numbers, as the coefficients)

(m) For every nonzero complex number, there is a unique complex number such that their product is π. (Real numbers, as the coefficients)

1.3.2 For each of the propositions in exercise I, write a useful denial, and give a translation into ordinary English. (Solution not available yet; Solve it!)

1.3.3 Give two proofs of Theorem 1.3(b). (Solution not available yet; Solve it!)

Theorem 1.3(b): If A (x) is an open sentence with variable x, then ¬(∃x)A(x) is equivalent to (∀x)¬A(x).

1.3.4 Which of the following are true? The universe for each is given in parentheses. (Solution not available yet; Solve it!)

(a) (∀x)(x + x ≥ x) (Real numbers)

(b) (∀x)(x + x ≥ x) (Natural numbers)

(c) (∃x)(2x + 3 = 6x + 7) (Natural numbers)

(d) (∃x)(3^x = x^2 ) (Real numbers)

(e) (∃x)(3^x = x) (Real numbers)

(f) (∃x)(3(2 – x) = 5 + 8(1 – x)) (Real numbers)

(g) (∀x)(x^2 + 6x + 5 ≥ 0) (Real numbers)

(h) (∀x)(x^2 + 4x +5 ≥ 0) (Real numbers)

(i) (∃x)(x^2 + x + 41 is prime) (Natural numbers)

(j) (∀x)(x^2 + x + 41 is prime) (Natural numbers)

(k) (∀x)(x^3 + 17x^2 + 6x + 100 ≥ 0) (Real numbers)

1.3.5 Give an English translation for each. The universe is given in parentheses. (Solution not available yet; Solve it!)

(a) (∀x)(x ≥ I) (Natural numbers)

(b) (∃!x)(x ≥ 0 ∧ x ≤ 0) (Real numbers)

(c) (∀x)( x is prime ∧ x ≠ 2 ⇒ x is odd) (Natural numbers)

(d) (∃!x)(ln(x) = 1) (Real numbers)

(e) ¬(∃x)(x^2 < 0 ) (Real numbers)

(f) (∃!x)(x^2 = 0) (Real numbers)

(g) (∀x)(x is odd ⇒ x^2 is odd) (Natural numbers)

1.3.6 Which of the following are true for the universe of all real numbers? (Solution not available yet; Solve it!)

(a) (∀x)(∃y)(x + y = 0).

(b) (∃x)(∀y)(x + y = 0).

(c) (∃x)(∃y)(x^2 + y^2 = -1)

(d) (∀x)[x > 0 ⇒ (∃y)(y < 0 ∧ xy > 0)]

(e) (∀y)(∃x)(∀z)(xy = xz)

(f) (∃x)(∀y)(x ≤ y)

(g) (∀y)(∃x)(x ≤ y)

(h) (∃!y)(y < 0 ∧ y + 3 > 0)

(i) (∃!x)(∀y)(x = y^2)

(j) (∀y)(∃!x)(x = y^2)

(k) (∃!x)(∃!y)(∀w)(w^2 > x – y)

1.3.7 (a) Give a proof of (∃!x)P(x) ⇒ (∃x)P(x). (b) Show that the converse of the conditional sentence in (a) is false. (Solution not available yet; Solve it!)

1.3.8 Write a symbolic translation of the Mean Value Theorem from calculus. (Solution not available yet; Solve it!)

1.3.9 Write a symbolic translation of the definition of limf(x) = L, x->a. Find a useful denial and give an idiomatic English version. (Solution not available yet; Solve it!)

1.3.10 Which of the following are denials of (∃!x)P(x)? (Solution not available yet; Solve it!)

(a) (∀x)(P(x))∨(∀x)(¬P(x))

(b) (∀x)(¬P(x))∨(∃y)(∃z)(y ≠ z ∧ P(y) ∧ P(z))

(c) (∀x)[P(x) ⇒ (∃y)(P(y) ∧ x ≠ y)]

(d) ¬(∀x)(∀y)[(P(x) ∧ P(y)) ⇒ x = y]

1.3.11 Give a denial of “You can fool some of the people all of the time and all of the people some of the time, but you cannot fool all of the people all of the time.” (Solution not available yet; Solve it!)

1.3.12 Riddle: What is the English translation of the symbolic statement ∀∃∃∀? (Solution not available yet; Solve it!)

*All course documents available here.*

- 1.2.2 Write the converse and contrapositive of each conditional sentence in exercise 1

(Solution not available yet; Solve it!)

- 1.2.3 Identify the antecedent and consequent for each conditional sentence in the following statements from this book.

(Solution not available yet; Solve it!)

- 1.2.6 Make truth tables for these propositional forms.

(Solution not available yet; Solve it!)

- 1.2.7 Prove Theorem 1.2 by constructing truth tables for each equivalence.

(Solution not available yet; Solve it!)

- 1.2.8 Rewrite each of the following sentences using logical connectives. Assume that each symbol f, n, x, S, B represents some fixed object.

(Solution not available yet; Solve it!)

- 1.2.9 Show that the following pairs of statements are equivalent.

(Solution not available yet; Solve it!)

- 1.2.10 Give, if possible, an example of a true conditional sentence for which

(Solution not available yet; Solve it!)

- 1.2.11 Give, if possible, an example of a false conditional sentence for which

(Solution not available yet; Solve it!)

- 1.2.12 Give the converse and contrapositive of each sentence of exercise 8(a), (b), (c), and (d). Tell whether each converse and contrapositive is true or false.

(Solution not available yet; Solve it!)

- 1.2.13 The inverse, or opposite, of the conditional sentence P?Q is ¬P?¬Q.

(Solution not available yet; Solve it!)

- 1.2.14 Determine whether each of the following is a tautology, a contradiction, or neither.

(Solution not available yet; Solve it!)

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thisismath

vimeo

Also, I have updated the problems to include a link to the correct solutions.

I will be posing problems from section 1.2 today, but problems 1.1.1, 1.1.2, 1.1.6, 1.1.8, and 1.1.10 still do not have correct solutions!

Finally, thank you to everyone that is watching and especially those who are going through the problems. I know that this takes time out of your day, and I hope that I am making that time a valuable investment in your mathematical knowledge. Let me know if there is anything you think I could do better–even if you have told me it already.

Rick Hanlon II

]]>*All course documents available here.*

- 1.2.2 Write the converse and contrapositive of each conditional sentence in exercise 1

(Solution not available yet; Solve it!)

- 1.2.3 Identify the antecedent and consequent for each conditional sentence in the following statements from this book.

(Solution not available yet; Solve it!)

- 1.2.6 Make truth tables for these propositional forms.

(Solution not available yet; Solve it!)

- 1.2.7 Prove Theorem 1.2 by constructing truth tables for each equivalence.

(Solution not available yet; Solve it!)

- 1.2.8 Rewrite each of the following sentences using logical connectives. Assume that each symbol f, n, x, S, B represents some fixed object.

(Solution not available yet; Solve it!)

- 1.2.9 Show that the following pairs of statements are equivalent.

(Solution not available yet; Solve it!)

- 1.2.10 Give, if possible, an example of a true conditional sentence for which

(Solution not available yet; Solve it!)

- 1.2.11 Give, if possible, an example of a false conditional sentence for which

(Solution not available yet; Solve it!)

- 1.2.12 Give the converse and contrapositive of each sentence of exercise 8(a), (b), (c), and (d). Tell whether each converse and contrapositive is true or false.

(Solution not available yet; Solve it!)

- 1.2.13 The inverse, or opposite, of the conditional sentence P?Q is ¬P?¬Q.

(Solution not available yet; Solve it!)

- 1.2.14 Determine whether each of the following is a tautology, a contradiction, or neither.

(Solution not available yet; Solve it!)

]]>

*All course documents available here.*

*All course documents available here.*

*All course documents available here.*

Approximate transcript (Available here):

** **

**Fundamentals of Advanced Mathematics**

** Lesson 0: Introduction**

Understanding mathematics is a prerequisite to all other understanding.

This is because, unlike other forms of scientific understanding which use inductive reasoning, mathematics uses deductive reasoning.

**Inductive reasoning: **moves from specific premise to general conclusion.

**Deductive reasoning:** moves from general premise to a specific conclusion.

That is, generally natural and social scientists make observations of particular cases or phenomena and seek a general theory that describes or explains them. Mathematicians, on the other hand, use logic to draw conclusions based on the statements already proven as true.

Now, every mathematical principle is based off axioms.

**Axioms:** statements or propositions that are regarded as being established, accepted, or self-evidently true.

For example,

1 is the smallest number

two sets are equal only if they have the same elements

two parallel lines never intersect.

Now, for example, if you were to say that this last axiom was false, that would give you the non-eucledian geometry that you would study in a modern geometry course.

Mathematicians take axioms and the conclusions that necessarily follow from them to build mathematical theories. Thus the conclusions of a mathematician are proven to be true provided the assumptions are true. If the results of a mathematical theory are deemed incompatible with some portion of reality, the fault is not in the theory but with the assumptions about reality that make the theory inapplicable to that portion of reality.

E.G.

Let’s take as an axiom that all men are mortal. This is obvious, no one has ever heard of or seen anyone immortal, but we cannot prove it logically.

Let’s also take as an axiom that Socrates is a man. Again, this is certainly obvious but is in no way logically prove-able.

However, given that all men are mortal and that Socrates is indeed himself a man then we can make a deductive argument and say that Socrates must also be mortal. That is, we have proven that he is mortal based on our two axioms. If it is later shown that Socrates is some sort of immortal entity, then the error is not in our argument that he is a mortal, but in our axioms. Either all men are not mortal, or Socrates is not a man.

This course will teach you to think like a mathematician; to think deductively. There is no prerequisite. Three types of people will take this course:

One: people with absolutely no mathematics background past the mandatory high school algebra. Though this may give you problems in other high level mathematics courses, that is not a problem here. Indeed, this may even be a benefit; you will be able to study the language of advanced maths without being tainted by computational drudgery and formula memorization.

Two: people who have taken elementary college level courses, typically including calculus differential equations and linear algebra, and are interested in expanding their mathematical knowledge. This course may feel a little different than any other math course you have taken. Up to this point, your courses were probably based on systematically solving problems, and thus emphasis was placed on the computational tools and algorithms for this effect. In more advanced mathematics, however, the emphasis is placed on concepts and their connections; this requires intense thought, intuition, and understanding of the objects at hand. Thus you may have to forget everything you thought you knew about math and re-compile your understanding. This is a benefit, because now you can own the understanding.

People already acquainted with advanced maths. This course will be an easy overview for them and they will be able to contribute content and commentary for everyone else.

When it comes down to it, you simply cannot learn advanced maths by watching a video. If that is your goal, thanks for watching, quit now. Learning advanced maths requires a determination to really understand the topics covered. It requires accepting nothing anyone tells you as truth until you have convinced yourself of it’s validity. That is you have to own the understanding. Make it yours. You should be able to say ‘I understand the difference between for all there exists and there exists for all’ not because you memorized the difference but because you sorted through the logical difference on your own. For this reason I strongly encourage you to spend more time in the problems section then the video section. These are not the type of systematic problems you’re used to. In general you will be asked to prove something based on something else. The most important aspect here is the act of drawing the logical conclusions on your own. Repeating this process is what will reshape the way you think and the way you interpret the world around you. This is the lasting value of studying mathematics, everything else is insignificant in comparison.

After you have thoroughly digested a problem, share you solution with others; talk about how you arrived at it and help others who are struggling. Remember though, you are wasting your time if you simply look up the answer before thoroughly considering the problem.

I am not here to teach you in the sense of presenting knowledge for the purpose of memorization. I am merely a guide. I will present you the language and topics but it is your responsibility to own the understanding–to sort through the logic to understand fundamentally what is going on.

Thanks for watching, This is Math.

]]>I just posted lesson 0!

I decided to make it a page, opposed to a post, because of the full-width option.

Tomorrow I will finish adapting a post template to be full page so that comments can be posted on the lesson post.

For now, you can comment on the new lesson here!

**EDIT: Updated, you can find the new Lesson 0 post here.**

Thank you to the users who have already registered on this site, and don’t forget to register on University of Reddit.

Right now I am ironing out any of the major design aesthetics needed for the site to run smoothly. I spent some time trying to get the facebook connect to work so you guys could login with facebook, but it doesn’t look like that is going to happen for now–which is an excellent segway into my next point: I need help on a few items.

Apart from feedback and moderation, I need some advice on how to implement a problems/solutions section of the site. Currently I am considering making a subreddit for courses as we go to satisfy this need. The voting system there would make aggregating and commenting on solutions seamless, and a markup extension would be easily implemented.

I am also looking for some really smart people to come on board to hold me accountable mathematically, and if anyone has experiance working with WordPress I welcome your help.

If you’re interested, e-mail me: rickhanlonii@gmail.com

Stay tuned, I am working frantically to get everything in place to start lessons this weekend. Get pumped!

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