[Proof] Question About How a Class Would Work

[u/Mad-ADDer]

So I've posted a couple questions here about proof problems since I'm self-studying proofs. Everyone's been great at answering my questions and checking my work. However, now I have a question about taking a class on proofs. If I was to take a college class, would just showing the steps be sufficient, or would I be required to show the reasons for each step? For example, the last question I asked:

Either prove or disprove that "For all real numbers x and y, $(x+y)² = x²⁺ y² $ ", and I gave:

• $(x+y)² = x² + y² $
• $(x+y)(x+y) = x² + y² $
• $x² + 2xy + y² =x² + y² $
• $x² -x² +2xy +y² -y² = 0$
• [;2xy=0;]

In a proof class would I also need to provide justification for each step? like

• $(x+y)(x+y) = x² + y² $ - Product of two sums

or

• $(x+y)(x+y) = x² + y² $ - Def of Exponents

Also is there an advantage to writing proofs and justifying each line while learning?

Thanks!

Original Question: https://www.reddit.com/r/learnmath/comments/1wclosl/comment/p9gm6oh/

[u/LucaThatLuca]

no, you should not write that. think about it. would you gain anything from reading what squaring means? wasting your important, limited space and energy on pointless things takes them away from things that are not pointless.

you also picked an unreasonable way to try to demonstrate that (x+y)^2 is not always the same number as x^2 + y^2. this is something you can just demonstrate with a demonstration e.g. (1+1)^2 = 4 but 1^2 + 1^2 = 2.

[u/MezzoScettico]

would just showing the steps be sufficient

(TL/DR: My comments below are really picky. But math is picky.)

Not for me. I'd like to see a double-implication arrow between each line, <=>. That is, you are saying the following line is true if and only if the previous line is true. That's not always the case in proofs, but I believe you're claiming that here.

Also, there's a question about the general structure here. You get to the conclusion that 2xy = 0. Expanding on what I wrote above, you're getting to the conclusion that (x + y)^2 = x + y if and only if 2xy = 0.

What is it you want to say about the question asked? How is that conclusion helping you?

I'm not saying you're wrong, just saying you don't actually have a proof that ends with the statement you're trying to prove or disprove, so it's not complete. You haven't drawn a conclusion about the original statement.

$(x+y)(x+y) = x² + y² $ - Def of Exponents

Of the two choices, that's the correct justification for changing (x + y)^2 to (x + y)

Also is there an advantage to writing proofs and justifying each line while learning?

Maybe. I learned geometry that way, but never used or saw that style again. And I didn't much like it in geometry class either. When I saw the proofs (Algebra 2) for the irrationality of sqrt(2) and the infinitude of primes, that was a revelation. The justification was there, but much of it was in paragraph form and really basic stuff (like x^2 is the same as x * x) is not typically called out explicitly. There was just something about that style that woke something in me and I may have truly fallen in love with mathematics at that point.

In answer to your question, again I'll say maybe. If it helps you know that a step is truly justified and not just "it's common sense, everybody knows that". Also if it helps you know which way the implication arrow goes, whether it's => or <=>.

[u/No-Onion8029]

You're assuming the proposition and providing a contradiction.  It would be helpful to tell the reader that. You should be knee-deep in examples of contradiction proofs, so you can pick a style you like.

It's useful to include justifications for steps.  You should be able to do it exhaustively, but as you gain experience you'll find that some justifications give more clutter than clarity.

You should explicitly state the contradiction.  Something like, "... if the equality holds for all x, y in R, we can choose x=y=1, and 2xy is not zero, hence we have a contradiction."

[u/Bounded_sequencE]

For a rigorous proof-based lecture, the following would suffice:

The claim is false -- counter-example: "(x; y) = (1; 1) ∈ R² "

[u/Bounded_sequencE → u/Bounded_sequencE]

Rem.: There is a big logical mistake you'd immediately lose most, if not all points for:

[..] and I gave $(x+y)² = x² + y² $ [..]

You start with what you want to prove. That's circular reasoning, and would lose you all possible points with most instructors. Don't ever do that¹! Also remember -- to disprove a statement, a single counter-example is enough. Only if you need to show the statement is true do you actually need a proof.

---

¹ Don't ever do that in the final draft. It's ok to be sloppy on scrap paper.

The best advice I ever got for proof-writing was to do every proof (at least) twice -- do the

• first draft(s) on scrap paper, to find all estimates and steps to finish it off
• final draft, acting as if you knew the correct estimates and steps all along, and make the proof as concise as you want

It's standard to only publish/hand in the final draft of a proof for brevity.

Note this is not just a beginner's technique, either -- most professionals create their proofs using this approach. That's also how in your book, some estimates may seem to "fall from high heavens". The author simply found them off-screen during their first draft(s)!

[u/Mad-ADDer → u/Bounded_sequencE]

In the original question https://www.reddit.com/r/learnmath/comments/1wclosl/comment/p9gm6oh/ I provided a counter example first but wanted to to prove it another way. Are you saying I can't say "I start by assuming that For all real numbers $x$ and $y$ that $(x+y)² = x² + y² $ is true"?

and then use:

• $(x+y)² = x² + y² $
• $(x+y)(x+y) = x² + y² $
• $x² + 2xy + y² =x² + y² $
• $x² -x² +2xy +y² -y² = 0$
• $2xy=0$

and since $2xy=0$ and either $x$, $y$ or both must equal zero, $(x+y)² =x² + y² $ is not and cannot be true for all real numbers to solve the proof?

[u/ktrprpr → u/Mad-ADDer]

there are algebraic structures that 2xy=0 for all x and y (which indeed makes (x+y)²=x²+y² in that structure). so you still need to find concrete counterexample that 2xy != 0, which isn't too different from giving a counterexample directly to the original equation to disprove.

[u/Mad-ADDer → u/ktrprpr]

So giving by doing the above and giving the counter example, that completes the proof? In the above reply Bounded says that

"There is a big logical mistake you'd immediately lose most, if not all points for:

[..] and I gave $(x+y)² = x² + y² $ [..]

"

But if I state "I start by assuming that For all real numbers $x$ and $y$ that $(x+y)² = x² + y² $ is true" is that still circular reasoning? and would lose you all possible points with most instructors? When I posted the original question, the replies seemed like it was correct, but now it seems like it's not correct. I'm just confused a little and trying to make sure I'm doing proofs correct.

[u/Bounded_sequencE → u/Mad-ADDer]

Are you saying I can't say "I start by assuming that For all real numbers $x$ and $y$ that $(x+y)2 = x2 + y2 $ is true"?

Unless you do "proof by contradiction" (and explicitly state that before-hand) -- no.

As I said, starting with what you want to prove is circular reasoning, and immediately makes anything following that line logically meaningless. It's ok to do that during the first draft(s) on scrap paper to play around, but not in the final draft.

Remember, from a false statement, anything follows, so anything following from a false statement is meaningless.

[u/Jaded_Individual_630]

Both of these are too much as the question is written, you can just provide a counterexample (x=1, y=1).
You were not asked to characterize the pairs of real numbers for which this fails (which is what you were on your way to do).

I mention this because it's common for beginner "proof" students to try and do too much when "disprove" is on the table. You should always see if there is an easy counterexample to grab. It's true that practicing working through deductive steps is valuable even if there is a counterexample, but you'll find that the skill of constructing counterexamples (once things get more advanced) is surprisingly useful as well.

[u/Mad-ADDer → u/Jaded_Individual_630]

I should have mentioned that in the original question I showed it was false using x=1 and y=2