Reddit Reddit reviews How to Read and Do Proofs: An Introduction to Mathematical Thought Processes

We found 6 Reddit comments about How to Read and Do Proofs: An Introduction to Mathematical Thought Processes. Here are the top ones, ranked by their Reddit score.

Science & Math
Books
Mathematics
Mathematical Logic
Pure Mathematics
How to Read and Do Proofs: An Introduction to Mathematical Thought Processes
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6 Reddit comments about How to Read and Do Proofs: An Introduction to Mathematical Thought Processes:

u/GeneralEbisu · 6 pointsr/math

I'm also planning on doing a Masters in Math or CS. What do you plan to write for your masters?


> Anybody else feels like this?

I think its natural to doubt yourself, sometimes. I dont know what else to say, but just try to be objective and emotionless about it (when you get stuck in a problem).

The following books that helped me improve my math problem solving skills when I was an undergrad:

u/let_me_count_the_way · 4 pointsr/HomeworkHelp

What this expressions says

First of all let's specify that the domain over which these statements operate is the set of all people say.
Let us give the two place predicate P(x,y) a concrete meaning. Let us say that P(x,y) signifies the relation x loves y.

This allows us to translate the statement:
∀x∀yP(x,y) -> ∀xP(x,x)

What does ∀x∀yP(x,y) mean?

This is saying that For all x, it is the case that For all y, x loves y.
So you can interpret it as saying something like everyone loves everyone.

What does ∀xP(x,x) mean?

This is saying that For all x it is the case that x loves x. So you can interpret this as saying something like everyone loves themselves.

So the statement is basically saying:
Given that it is the case that Everyone loves Everyone, this implies that everyone loves themselves.
This translation gives us the impression that the statement is true. But how to prove it?

Proof by contradiction

We can prove this statement with a technique called proof by contradiction. That is, let us assume that the conclusion is false, and show that this leads to a contradiction, which implies that the conclusion must be true.

So let's assume:
∀x∀yP(x,y) -> not ∀xP(x,x)

not ∀xP(x,x) is equivalent to ∃x not P(x,x).
In words this means It is not the case that For all x P(x,x) is true, is equivalent to saying there exists x such P(x,x) is false.

So let's instantiate this expression with something from the domain, let's call it a. Basically let's pick a person for whom we are saying a loves a is false.

not P(a,a)

Using the fact that ∀x∀yP(x,y) we can show a contradiction exists.

Let's instantiate the expression with the object a we have used previously (as a For all statement applies to all objects by definition) ∀x∀yP(x,y)

This happens in two stages:

First we instantiate y
∀xP(x,a)

Then we instantiate x
P(a,a)

The statements P(a,a) and not P(a,a) are contradictory, therefore we have shown that the statement:

∀x∀yP(x,y) -> not ∀xP(x,x) leads to a contradiction, which implies that
∀x∀yP(x,y) -> ∀xP(x,x) is true.

Hopefully that makes sense.

Recommended Resources

Wilfred Hodges - Logic

Peter Smith - An Introduction to Formal Logic

Chiswell and Hodges - Mathematical Logic

Velleman - How to Prove It

Solow - How to Read and Do Proofs

Chartand, Polimeni and Zhang - Mathematical Proofs: A Transition to Advanced Mathematics

u/LADataJunkie · 3 pointsr/ucla

You will want to jump on 115A, but have a back up class in case you need to drop and realize it isn't going to work. I dropped 115A twice before I could finally commit and feel mature enough to do well in it.

One thing that really helped me was taking Combinatorics, a field that is fascinating to me. There was *some* proof writing in the class, but it was pretty basic (similar to proofs in statistics). I enjoyed writing those proofs and taught me the entire purpose of doing it. I was then able to do 115A with little difficulty.

I also got the following book, which is excellent (I used a much older edition) How to Read and Do Proofs by Daniel Solow.

https://www.amazon.com/How-Read-Proofs-Introduction-Mathematical/dp/1118164024/ref=pd_sbs_14_1?_encoding=UTF8&pd_rd_i=1118164024&pd_rd_r=DRKKPQHM9KM7NF7XS7AJ&pd_rd_w=jAI7z&pd_rd_wg=TfejV&psc=1&refRID=DRKKPQHM9KM7NF7XS7AJ&dpID=51ljxm2YBEL&preST=_SY344_BO1,204,203,200_QL70_&dpSrc=detail

u/funnythingaboutmybak · 2 pointsr/learnmath

I got my bachelors in Spanish and I have one more semester left to finish my masters in mathematics. As someone coming from a liberal arts degree, proofs were foreign to me, and handling anything with more than one variable was just asking too much. When I took Linear Algebra (my first proofs class), I had peers like the ones you mentioned, who just "got it" without taking notes while missing a third of the class lectures. And here I was slaving away, lost in a web of confusion. That class almost broke me. But over time, I learned a few things which were catalysts for my math competency:

  • I learned the framework of proofs and logic from this awesome book so that whenever I saw a theorem, I already had an idea of how to tackle it
  • I gave no shits about people thinking I'm stupid and thus asked A LOT of questions in class
  • I showed up to office hours religiously; it's insane not to utilize this one-on-one time with a PhD with tuition rates being as they are
  • I drew everything I could to help me understand concepts; if you can see the forest from the trees, the details will fall out
  • Related to the above, I tried building a visual intuition of things I was learning which helped me see past the slew of variables and greek letters
  • I memorized all definitions and stuck them in Anki; you're screwed trying to do proofs without definitions
  • Those other smart guys had to do everything I was doing at
    some point in time, they just got a head start; so I put in the hours and caught up (even surpassed in some areas)
  • I immersed myself in math: gave talks at conferences, got a job related to math, talked to other students about it, blogged about it, etc...

    Anyway, the struggle is real, but after all those focused hours of engaged studying, the intuition will finally be there and your brain will then "compress" that information so you have room to learn more. Ad infinitum.

    Hope that helps.
u/phlummox · 1 pointr/learnmath

Oh, I'm terrible at calculus, haha. I teach discrete maths and logic, and never have to touch calculus at all, thank goodness :)

But a younger friend of mine is doing calculus just now, so I'll find out what he found useful and PM you. He did say that some of the books I'd recommended him were immensely useful for maths generally (not necessarily calculus in particular). In roughly ascending order of difficulty: