This tag is for readers who ask for explanation and clarification of some steps of a particular proof.

# Questions tagged [proof-explanation]

10133 questions

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### The "assumption" in proof by induction

The second step in proof by induction is to:
Prove that if the statement is true for some integer $n=k$, where
$k\ge n_0$ then it is also true for the next larger integer, $n=k+1$
My question is about the "if"-statement. Can we just assume that…

GambitSquared

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### Irreducibles are prime in a UFD

Any irreducible element of a factorial ring $D$ is a prime element of
$D$.
Proof. Let $p$ be an arbitrary irreducible element of $ D$. Thus $ p$
is a non-unit. If $ ab \in (p)\smallsetminus\{0\}$, then $ ab = cp$
with $ c \in D$. We write $…

emmett

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### Confused by proof of the irrationality of root 2: if $p^2$ is divisible by $2$, then so is $p$.

In typical proofs of the irrationality of $\sqrt{2}$, I have seen the following logic:
If $p^2$ is divisible by $2$, then $p$ is divisible by $2$.
Perhaps I am being over-analytical, but how do we know this to be true? IE. do we require a proof of…

user322548

**21**

votes

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### Why do we do mathematical induction only for positive whole numbers?

After reading a question made here, I wanted to ask "Why do we do mathematical induction only for positive whole numbers?"
I know we usually start our mathematical induction by proving it works for $0,1$ because it is usually easiest, but why do we…

Simply Beautiful Art

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### Problems understanding proof of if $x + y = x + z$ then $y = z$ (Baby Rudin, Chapter 1, Proposition 1.14)

I'm having trouble with whether Rudin actually proves what he's tried to prove.
Proposition 1.14; (page 6)
The axioms of addition imply the following statements:
a) if $x + y = x + z$ then $y = z$
The author's proof is as follows:
$ y = (0 + y) = (x…

Brayton

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### Hartshorne Theorem 8.17

I can't understand the proof of theorem 8.17 from Hartshorne's "Algebraic Geometry". Namely, he says that we have an exact sequence
$$
0 \to \mathcal J'/\mathcal J'^2 \to \Omega_{X/k} \otimes \mathcal O_{Y'} \to \Omega_{Y'/k} \to 0
$$
and there…

qwerzzx

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### How do we know that the Sine function has no Non-Real Roots?

In this question and answers (How was Euler able to create an infinite product for sinc by using its roots?) we use the fact that the real roots of $f(x)=\sin x$ occur when $x$ is an integer multiple of $\pi$ to obtain an infinite product for $\sin…

A-Level Student

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### Decomposition of a representation into a direct sum of irreducible ones

I'm studying representation theory and in the book (Fulton and Harris) the author makes the following proposition with the following proof:
Proposition: For any representation $V$ of a finite group $G$, there is a decomposition
$$V = V_1^{\oplus…

Gold

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### A Proof with no words that $\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}=2$

Question
What are the words to describe the method in the image below? (from Nelsen's Proofs without Words II)
Attempt
I was thinking and could define the sequence $u_1=2; u_{n+1}=f\circ g^{−1}(u_n)$ where $f(x)=\sqrt x$ and $g(x)=x−2$, as…

Pedro Costa

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### Why does proof by contrapositive make intuitive sense?

If you have two statements P and Q, and we say that P implies Q, that suggests that P contains Q. So if we have P, we must have Q because it is contained within P. This is my intuitive understanding of the implication.
On the other hand, if we do…

IgnorantCuriosity

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### What's going on with this 5-line proof of Fermat's Last Theorem?

I'm reading a book on the Philosophy of Mathematics, and the author gave a "5-line proof" of Fermat's Last Theorem as a way to introduce the topic of inconsistency in set theory and logic. The author acknowledges that this is not a real proof of the…

IgnorantCuriosity

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### Nonsense from combining two iffs ($\iff$)

E and F are independent iff $\frac{p(E \cap F)}{p(F)}=p(E)$.
Also, E and F are independent iff $p(E | F)=p(E)$
Why do I get nonsense if I combine the two? I'd get: E and F are independent iff $p(E | F) = \frac{p(E \cap F)}{p(F)}$
Why doesn't the…

JobHunter69

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### Help understand the proof of infinitely many primes of the form $4n+3$

This is the proof from the book:
Theorem. There are infinitely many primes of the form $4n+3$.
Lemma. If $a$ and $b$ are integers, both of the form $4n + 1$, then the product $ab$ is also in this form.
Proof of Theorem:
Let assume that there are…

Chan

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### How to convert $\pi$ to base 16?

According to this Wikipedia article $\pi$ is approximately 3.243F in base 16 (i.e. hexadecimal).
Can someone explain this? (Note: I understand how to convert an integer to base 16)
Thanks

Ben

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### Is it true that $\int_0^\infty \frac{f(x)}{x} \sin \bigl(\frac{\pi x}{a}\bigr) \,\mathrm{d}x = \frac{\pi}{2} \int_0^{a/2} f(x) \,\mathrm{d}x$?

I dont recall where, but I found this interesting identity a few years ago. It was shown in one of Victor Moll's papers about elliptic integrals.
Corollary 3.1. Let $f$ be an even function with period $a$. Then,
$$\int_0^\infty \frac{f(x)}{x}…

N3buchadnezzar

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